Correction: The early Church and Eastern Churches confer Baptism, Confirmation, and Eucharist together, to infants, and to converting families.
"The early Church and Eastern Churches confer Baptism, Confirmation, and Eucharist together, to infants, and to converting families."
Right you are as to infants and as to those who have not been baptised by water and in the name of the Father, the Son and the Holy Spirit. For baptised converts from Roman Catholicism or one of the protestant "ecclesial assemblies", its Chrismation (Confirmation) and the Eucharist.