Markov Chain w/ non-stochastic matrix
No, there's no standard name for a process where the transpose of the transition matrix is stochastic (at least, to my knowledge).

However, this process is the time-reversal of a Markov process.  That's a remarkable and descriptive property that just about completely characterizes this process.  In particular, it seems likely that many things we'd want to know about such a process can be derived from the fact that it's the time-reversal of a Markov process, plus known facts about Markov processes.

Anyway, if you need to describe the process, "time-reversal of a Markov process process" is probably about as good as you're going to get.