Let $X_{1},X_{2}$ be discrete topological spaces and $p_{1},p_{2}$ two transitions functions corresponding to random walks on $X_{1}$ and $X_{2}$, respectively. If $f:X_{1}\rightarrow X_{2}$ is a continuous map, are there conditions one could put on $f$ to say it "maps" the random walk for $p_{1}$ to the random walk for $p_{2}$?
Perhaps a more general question would be, is there a sensible definition of maps between random walks?