A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial states, the markov convergences to the same steady states with some known properties of $F(\cdot)$. I was trying to find some references about this issues and hardly find some useful ones. Can anyone provide some paper or whatever references addressing this kind of problem?

There might not be a steady state, in general. Mixing is the key notion that can still guarantee convergence in the non-homogeneous case. Here is one result out of many: http://arxiv.org/abs/0807.4665

See also this related question: Time-inhomogeneous Markov Chains