Notes for context: I am a native BrEng speaker. I have read https://english.stackexchange.com/questions/126726/each-of-these-is-vs-each-of-these-are, https://english.stackexchange.com/questions/24846/how-does-each-change-are-to-is, and https://english.stackexchange.com/questions/24777/what-should-i-use-between-triple-vs-all and I understand the general concept of "Each of them is" and how "each" always takes the singular. 

Reading my daughter's bedtime story this evening, I came across the phrase "They each have a bag of equipment". It made me wonder why the following seems naturally (to me at least) to be true:  
>(Correct) Each of them has an X  
>(Correct) They each have an X  
>(Correct) Each of them is a Y  
>(Incorrect?) They each are a Y  

I would never say (to use the examples from one of the posts I linked) "I have three pens. They each are green.", but "I have three pens. They each have a lid." is fine.

Is "to have" different from "to be" when it comes to using it with "each"?