We consider a nonlocal parabolic system with a singular target space. Caffarelli and Lin showed that a well-known optimal eigenvalue partition problem could be reformulated as a constrained harmonic mapping problem into a singular space. We show that the heat flow corresponding to this problem is Lipschitz continuous, and study the regularity of a resulting free interface. We also show that the flow converges to a stationary solution of the constrained mapping problem as time approaches infinity. Time permitting, we will also discuss some related ongoing work involving more general non-smooth target spaces.