  We introduce a general approach for the study of the collective dynamics of
non-interacting random walkers on connected networks. We analyze the movement
of $R$ independent (Markovian) walkers, each defined by its own transition
matrix. By using the eigenvalues and eigenvectors of the $R$ independent
transition matrices, we deduce analytical expressions for the collective
stationary distribution and the average number of steps needed by the random
walkers to start in a particular configuration and reach specific nodes the
first time (mean first-passage times), as well as global times that
characterize the global activity. We apply these results to the study of mean
first-encounter times for local and non-local random walk strategies on
different types of networks, with both synchronous and asynchronous motion.
