  We analyze the effect of adding, removing, and moving basepoints on link
Floer homology. We prove that adding or removing basepoints via a procedure
called quasi-stabilization is a natural operation on a certain version of link
Floer homology, which we call $CFL_{UV}^\infty$. We consider the effect on the
full link Floer complex of moving basepoints, and develop a simple calculus for
moving basepoints on the link Floer complexes. We apply it to compute the
effect of several diffeomorphisms corresponding to moving basepoints. Using
these techniques we prove a conjecture of Sarkar about the map on the full link
Floer complex induced by a finger move along a link component.
