I don't think they are completly "coming home" --- more being in communion, with acceptable doctrinal differences.
Although it is not widely known in our Western world, the Catholic Church is actually a communion of Churches. According to the Constitution on the Church of the Second Vatican Council, Lumen Gentium, the Catholic Church is understood to be "a corporate body of Churches," united with the Pope of Rome, who serves as the guardian of unity (LG, no. 23). At present there are 22 Churches that comprise the Catholic Church. The new Code of Canon Law, promulgated by Pope John Paul II, uses the phrase "autonomous ritual Churches" to describe these various Churches (canon 112). Each Church has its own hierarchy, spirituality, and theological perspective.
However, they all share the same doctrines of faith.
Incorrect. You obviously don't know what the meaning of "to come into full communion with the Holy See." actually means.