A general technique is given for constructing new Markov processes from existing ones. The new process and its state space are both projective limits of sequences built by an iterative scheme. The space at each stage in the scheme is obtained by taking disjoint copies of the space at the previous stage and quotienting to identify certain distinguished points. Away from the distinguished points, the process at each stage evolves like the one constructed at the previous stage on some copy of the previous state space, but when the process hits a distinguished point it enters at random another of the copies ``pinned'' at that point. Special cases of this construction produce diffusions on fractal-like objects that have been studied recently.