  We consider a family of nonlocal curvatures determined through a kernel which
is symmetric and bounded from above by a radial and radially non-increasing
profile. It turns out that such definition encompasses various variants of
nonlocal curvatures that have already appeared in the literature, including
fractional curvature and anisotropic fractional curvature. The main task
undertaken in the article is to study the limit behaviour of the introduced
nonlocal curvatures under an appropriate limiting procedure. This enables us to
recover known asymptotic results e.g. for fractional curvature, but also for
anisotropic fractional curvature where we identify the limit object as a
curvature being the first variation of the related anisotropic perimeter. Our
other goal is to prove existence, uniqueness and stability of viscosity
solutions to the corresponding level-set parabolic Cauchy problem formulated in
terms of the investigated nonlocal curvature.
