Given a differentiable manifold $M$, a compact subset $U ⊆ M$, a finite subset $C ⊆ U$ and an injective function $f_C: C → U$, when does there exist a diffeomorphism $f: M → M$ that is "supported" on $U$ (i.e. $f(m) = m$ if $m ∉ U$), such that $f(c) = f_C(c)$ for all $c ∈ C$? Is there a construction for $f$? My specific focus is on $C^∞$ manifolds, though useful results might exist for $C^n$ manifolds for positive integers $n$.
I have no idea where to find any results like this, because I don't even know what you would name the important objects underlying these statements or required for the results, other than the few that were named. This bears similarity to the "morphing" problem in image graphics.
Edit: as a shot in the dark, I asked a certain AI assistant with a well-known name and it said to look at the "Whitney Approximation Theorem", which it claims matches the request, item by item; and it suggested the obvious tools for construction: using locally finite cover by coordinate charts + partitions of unity.
But an "hallucination check" on the AI assistant shows that though the theorem name is a thing, its formulation apparently involves smoothing out $C^n$ functions, for finite $n$, which is weaker than trying to match a finite point cloud transform. The reduction to coordinate charts is not straightforward, in either case, but especially not for point clouds, because the transforms of the points may lie in totally separate areas on different charts.