Step One: who or what are "The Eastern Churches"?!
Most Catholics cannot answer this question.
This article has an excellent explanation

On reading the article, one cannot help but ask why such a rich variety of rites cannot be expanded to include other Christians? 

We now have an Anglican "Ordinariate"....not unlike the arrangement for the Eastern Churchs.
May there someday soon be  Baptist/Presbyterian/Uniting Church etc Ordinariates in the Church

Come Lord Jesus, Come!