Let $S_g$ be a closed and oriented surface of genus $g\ge 2$.
Given any two finite subsets $P_1,P_2\subset S_g$ of points of the same cardinality and any bijection $\phi:P_1\rightarrow P_2$, I am wondering if it is always possible to find a diffeomorphism $f:S_g\rightarrow S_g$ such that:
$f_*:\pi_1(S_g)\rightarrow \pi_1(S_g)$ is the identity
$f|_{P_1}:P_1\rightarrow P_2$ is equal to $\phi$
If the answer to the general case is "no", then:
is it "yes" for sets $P_1,P_2$ of low cardinality?
is it "yes" if $p$ and $\phi(p)$ are sufficiently close to each other, for every $p\in P_1$?
Yes, you can find a map isotopic to the identity that sends any point $P$ to any other point $Q$ (assuming your manifold of dimension $\ge 2$ is connected). (Choose a smooth path from $P$ to $Q$ and choose a vector field tangent to that path that is compactly supported in a neighborhood of the path. Then flow along that vector field.) You can iterate this construction a finite number of times. (Make sure that neighborhood contains no other points from your list.)