It's maybe besides the point, but IMO if a T has something big going on in their life, and it is affecting them so much that they tear up in session, they're not capable of focusing on the client in that moment. It would be better to cancel the session entirely.

A couple times I've thought something seemed off with my T and asked him about it. He has usually said he's fine, or on two occasions admitted there was something going on (and I learned what it was later on) but quickly redirected the conversation back to me. I think it's perfectly natural to care about your T as a person, and yes, it's a relationship with two sides, but it's still meant to be for the client's benefit.