The mere existence of Eastern Catholic Churches, especially in the United States, is a form of evangelization. Many parishes were founded as remote mission churches from the mother land to serve immigrant populations, allowing them to keep their religious identity and avoid drifting away from the faith altogether.
In the mother land, there is an agreement not to actively encroach on the local Orthodox Church’s members, although this is a point of controversy in many places. Individual’s are welcome to enter the Catholic Church, but out of respect for the validity and licity of the Orthodox Churches, there is no active campaign for conversion. These souls are already in a church in imperfect union with Rome, and have access to the graces of the Sacraments.
The ECC’s are thus in a holding pattern. They were form a few hundred years ago in response to ecumenical talks between Rome and the Orthodox, in which some Orthodox entered communion with Rome, while the majority remained independent. While possessing the full rights of proper churches, they have limited membership, and many hope and pray to rejoin their Orthodox counterpart should a full reunion between Rome and greater Orthodoxy occur.