I think therapists are mainly responsible in maintaining boundaries, and would bring up contact between sessions. It's the lack of consistency that would bother me- why are some emails okay and some not?

One of my last ruptures with my T was about him ignoring an email. For me I've always needed to know that he was "there", not replying made me feel like I wasn't worth replying to and brought back stuff from when I was 14. He was defensive at first , but later apologized and said that he could see why it was important for me. I send one email over the weekend if I need to. So far he has been on the ball with it.