The Roman Catholic church's position on infant baptism has NOT been consistent, but has varied over time and depending upon which church "father" is followed at any one time. We can see that this nebulous view concerning where unbaptized babies who die spend eternity explained here http://www.newadvent.org/cathen/09256a.htm:
The New Testament contains no definite statement of a positive kind regarding the lot of those who die in original sin without being burdened with grievous personal guilt. But, by insisting on the absolute necessity of being "born again of water and the Holy Ghost" (John 3:5) for entry into the kingdom of Heaven (see BAPTISM, subtitle Necessity of Baptism), Christ clearly enough implies that men are born into this world in a state of sin, and St. Paul's teaching to the same effect is quite explicit (Romans 5:12 sqq.). On the other hand, it is clear from Scripture and Catholic tradition that the means of regeneration provided for this life do not remain available after death, so that those dying unregenerate are eternally excluded from the supernatural happiness of the beatific vision (John 9:4, Luke 12:40, 16:19 sqq., 2 Corinthians 5:10; see also APOCATASTASIS). The question therefore arises as to what, in the absence of a clear positive revelation on the subject, we ought in conformity with Catholic principles to believe regarding the eternal lot of such persons. Now it may confidently be said that, as the result of centuries of speculation on the subject, we ought to believe that these souls enjoy and will eternally enjoy a state of perfect natural happiness; and this is what Catholics usually mean when they speak of the limbus infantium, the "children's limbo." The best way of justifying the above statement is to give a brief sketch of the history of Catholic opinion on the subject. We shall try to do so by selecting the particular and pertinent facts from the general history of Catholic speculation regarding the Fall and original sin, but it is only right to observe that a fairly full knowledge of this general history is required for a proper appreciation of these facts.
Pre-Augustinian tradition
There is no evidence to prove that any Greek or Latin Father before St. Augustine ever taught that original sin of itself involved any severer penalty after death than exclusion from the beatific vision, and this, by the Greek Fathers at least, was always regarded as being strictly supernatural. Explicit references to the subject are rare, but for the Greek Fathers generally the statement of St. Gregory of Nazianzus may be taken as representative: It will happen, I believe . . . that those last mentioned [infants dying without baptism] will neither be admitted by the just judge to the glory of Heaven nor condemned to suffer punishment, since, though unsealed [by baptism], they are not wicked. . . . For from the fact that one does not merit punishment it does not follow that one is worthy of being honored, any more than it follows that one who is not worthy of a certain honor deserves on that account to be punished. [Oration 40, no. 23] Thus, according to Gregory, for children dying without baptism, and excluded for want of the "seal" from the "honor" or gratuitous favor of seeing God face to face, an intermediate or neutral state is admissible, which, unlike that of the personally wicked, is free from positive punishment. And, for the West, Tertullian opposes infant baptism on the ground that infants are innocent, while St. Ambrose explains that original sin is rather an inclination to evil than guilt in the strict sense, and that it need occasion no fear at the day of judgement; and the Ambrosiaster teaches that the "second death," which means condemnation to the hell of torment of the damned, is not incurred by Adam's sin, but by our own. This was undoubtedly the general tradition before St. Augustine's time.
Post-Augustinian teaching
After enjoying several centuries of undisputed supremacy, St. Augustine's teaching on original sin was first successfully challenged by St. Anselm (d. 1109), who maintained that it was not concupiscence, but the privation of original justice, that constituted the essence of the inherited sin (De conceptu virginali). On the special question, however, of the punishment of original sin after death, St. Anselm was at one with St. Augustine in holding that unbaptized children share in the positive sufferings of the damned; and Abelard was the first to rebel against the severity of the Augustinian tradition on this point. According to him there was no guilt (culpa), but only punishment (poena), in the proper notion of original sin; and although this doctrine was rightly condemned by the Council of Soissons in 1140, his teaching, which rejected material torment (poena sensus) and retained only the pain of loss (poena damni) as the eternal punishment of original sin (Comm. in Rom.), was not only not condemned but was generally accepted and improved upon by the Scholastics. Peter Lombard, the Master of the Sentences, popularized it (Sent. II, xxxiii, 5), and it acquired a certain degree of official authority from the letter of Innocent III to the Archbishop of Arles, which soon found its way into the "Corpus Juris". Pope Innocent's teaching is to the effect that those dying with only original sin on their souls will suffer "no other pain, whether from material fire or from the worm of conscience, except the pain of being deprived forever of the vision of God" (Corp. Juris, Decret. l. III, tit. xlii, c. iii — Majores). It should be noted, however, that this poena damni incurred for original sin implied, with Abelard and most of the early Scholastics, a certain degree of spiritual torment, and that St. Thomas was the first great teacher who broke away completely from the Augustinian tradition on this subject, and relying on the principle, derived through the Pseudo-Dionysius from the Greek Fathers, that human nature as such with all its powers and rights was unaffected by the Fall (quod naturalia manent integra), maintained, at least virtually, what the great majority of later Catholic theologians have expressly taught, that the limbus infantium is a place or state of perfect natural happiness. No reason can be given — so argued the Angelic Doctor — for exempting unbaptized children from the material torments of Hell (poena sensus) that does not hold good, even a fortiori, for exempting them also from internal spiritual suffering (poena damni in the subjective sense), since the latter in reality is the more grievous penalty, and is more opposed to the mitissima poena which St. Augustine was willing to admit (De Malo, V, art. iii). Hence he expressly denies that they suffer from any "interior affliction", in other words that they experience any pain of loss (nihil omnino dolebunt de carentia visionis divinae — "In Sent.", II, 33, q. ii, a. 2). At first ("In Sent.", loc. cit.), St. Thomas held this absence of subjective suffering to be compatible with a consciousness of objective loss or privation, the resignation of such souls to the ways of God's providence being so perfect that a knowledge of what they had lost through no fault of their own does not interfere with the full enjoyment of the natural goods they possess. Afterwards, however, he adopted the much simpler psychological explanation which denies that these souls have any knowledge of the supernatural destiny they have missed, this knowledge being itself supernatural, and as such not included in what is naturally due to the separated soul (De Malo loc. cit.).
It should be added that in St. Thomas' view the limbus infantium is not a mere negative state of immunity from suffering and sorrow, but a state of positive happiness in which the soul is united to God by a knowledge and love of him proportionate to nature's capacity. The teaching of St. Thomas was received in the schools, almost without opposition, down to the Reformation period. The very few theologians who, with Gregory of Rimini, stood out for the severe Augustinian view, were commonly designated by the opprobrious name of tortores infantium. Some writers, like Savonarola (De triumpho crucis, III, 9) and Catharinus (De statu parvulorum sine bapt. decedentium), added certain details to the current teaching — for example that the souls of unbaptized children will be united to glorious bodies at the Resurrection, and that the renovated earth of which St. Peter speaks (2 Peter 3:13) will be their happy dwelling place for eternity.
At the Reformation, Protestants generally, but more especially the Calvinists, in reviving Augustinian teaching, added to its original harshness, and the Jansenists followed on the same lines. This reacted in two ways on Catholic opinion, first by compelling attention to the true historical situation, which the Scholastics had understood very imperfectly, and second by stimulating an all-round opposition to Augustinian severity regarding the effects of original sin; and the immediate result was to set up two Catholic parties, one of whom either rejected St. Thomas to follow the authority of St. Augustine or vainly try to reconcile the two, while the other remained faithful to the Greek Fathers and St. Thomas. The latter party, after a fairly prolonged struggle, has certainly the balance of success on its side.
The Roman Catholic church today has all but tossed out even using the word "Limbo" and now the Catechism says:
In truth, we all should understand that God is gracious and merciful and no innocent human - infants, young children, mentally handicapped, etc. - who cannot, through no fault of his own, comprehend his sin and need of a savior, WILL be covered under the blood of Christ and will be in heaven with God at death. This is what God has revealed to us about His matchless grace through His sacred Scriptures.
You would be better off reading St. Thomas Aquinas or St. Augustine directly, rather than have someone else tell you what they said. In the Summa, Thomas does a good job of bringing in positions of the Early Church Fathers, ancient viewpoints, and the Sacred Scriptures. The City of God and Confessions by Augustine are good resources as well.
Thomas and Augustine are great theologians, and their understanding of sacred theology, the science which flows from divine revelation, is commendable. But it does not constitute “Church teaching”. Theologians can disagree, and put forth arguments that are later corrected by the Magisterium. Theologians also speculate, the concept of limbo is an example of such speculation.
Church practices in matters of discipline can vary over time. I can give you a recent example. When I was a child, the Byzantine Rite in America did not give Holy Communion to children under the age of reason. Over the last 40 years we have seen the Byzantine Rite adopt the practice of giving Holy Communion to every child regardless of age. It is a sweet tradition and we were happy to see it come about.
I hope this helps you. God bless you.