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@Pierre PC: Continuous dependence on prescribed boundary data follows from Caratheodory's kernel theorem, which gives conditions for local uniform convergence of conformal maps. See "Boundary Behavior of Conformal maps" by Pommerenke; specifically, Thm. 2.11 which Pommerenke attributes to Rado. This takes care of continuous dependence on a single boundary component. The case of multiple boundary components should be very similar, since the arguments are local.
@ Mikhail Katz: I don't know the best reference for CSF for polygons, but Carpenter's rule problem has been well studied in the paper of Connelly, Demaine and Rote.
@ Mikhail Katz: Curve shortening flow works for polygonal curves as well, and there are also other methods like the "Carpenter Rule Problem" which convexifies polygons in a canonical way.