  We study the fractional Schr\"odinger equation with quasilocal perturbations.
These are a family of nonlocal perturbations vanishing at infinity, which
include e.g. convolutions against Schwartz functions. We show that the
qualitative unique continuation and Runge approximation properties hold in the
assumption of sufficient decay. Quantitative versions of both results are also
obtained via a propagation of smallness analysis for the Caffarelli-Silvestre
extension. The results are then used to show uniqueness in the inverse problem
of retrieving a quasilocal perturbation from DN data under suitable geometric
assumptions. Our work generalizes recent results regarding the locally
perturbed fractional Calder\'on problem.
