The Assyrians and the Oriental Orthodox are not in communion with each other. The Assyrians reject accept only the first 2 ecumenical councils, while the Orientals accept only the first 3. They probably do share some traditions, but are definitely distinct bodies. Both, however, are recognized by the Catholic Church as Apostolic Churches with valid sacraments.
As to whether or not there is constant breaking and restoring of communion between the Oriental Churches, I could not say for sure. I know that currently there is some dispute with one of the Oriental Churches in India, but I don’t know enough about the situation to comment. The interesting thing about the Oriental Communion from a Catholic point of view (or at least from my point of view) is that they seem to differ in opinion concerning Catholics. The Armenians and Syriacs (as well as the Assyrians) tend to allow intermarriage with Catholics and seem to recognize our sacraments. The Copts, on the other hand, do not allow intermarriage with Catholics and have mixed opinions about our sacraments. I remember not too long ago a story about a Coptic bishop stating that Catholics could not be saved. I imagine that mardukm and Ghosty probably know a little more about Coptic-Catholic relations and the general OO Communion than I do, so hopefully they’'ll stop by at some point.