Let S be a compact orientable surface. Let A and B be two subsurfaces of S that have the same signature. How to check if there is a homeomorphism of S that sends A to B and if so, find one?

Here a subsurface of $S$ is a component of S \ simple closed curves.

I saw this question Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface and https://arxiv.org/pdf/math/9906122.pdf but I didn't find the part related to my question yet.