Notably, Flanders's (1970) interaction categories are divided into two types; namely, direct influences and indirect influences. The issue of empathy is generally related to indirect influences where L2 tearchers' response to learners' emotions and feelings has a great bearing on learning and teaching processes. Accordingly, research regarding conversational analysis addressing emotional management (empathy) is of a great empirical significance. The following links can hopefully provide you with the targeted literature you are looking for.
I am not exactly sure if empathy as a psychological notion is conversation analytic. To me, it would make sense to approach this issue through affiliative moves, and also through looking into references from other areas such as Mahmood above suggests.
Another interesting way of looking into empathy is through examining mimetic behaviour - which is actually not very far from CA thinking (although I do not think that mimesis is on the CA plate at all). Here comes a relevant work I've written:
Article The pragmatics of mimesis: A case study of intercultural ema...