I will discuss how (generalised) deformations of a smooth projective variety interact with arbitrary Fourier-Mukai functors (based on work by Toda). This machinery can be applied to P-functors, allowing one to obtain a criterion for a P-functor to become spherical on the total space of a deformation of the target, generalising a result by Huybrechts and Thomas. Finally, I will explain how this abstract criterion can be checked in the case of Hilbert schemes of points on a K3 surface, thus providing new examples of derived autoequivalences for certain deformations of these Hilbert schemes. This talk contains joint work (in progress) with Ciaran Meachan.