We study sequences (both cyclic and randomized) of idempotent completely positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods, highlighting the robustness features of the protocol against randomization. The general results are then specialized to stabilizing entangled states in finite-dimensional multipartite quantum systems subject to locality constraints, a problem of key interest for quantum information applications.