Small modifications of Mori dream spaces arising from C*-actions

We link small modifications of projective varieties with a C*-action to their GIT quotients. Namely, using flips with centers in closures of Bialynicki-Birula cells, we produce a system of birational equivariant modifications of the original variety, which includes those on which a quotient map extends from a set of semistable points to a regular morphism. The structure of the modifications is completely described for the blowup along the sink and the source of smooth varieties with Picard number one with a C*-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a C*-action associated to short grading of their Lie algebras.