  The paper provides two versions of nonlocal Poincar\'e-type inequalities for
integral operators with a convolution-type structure and functions satisfying a
zero-Dirichlet like condition. The inequalities extend existing results to a
large class of nonhomogeneous kernels with supports that can vary
discontinuously and need not contain a common set throughout the domain. The
measure of the supports may even vanish allowing the zero-Dirichlet condition
to be imposed on only a lower-dimensional manifold, with or without boundary.
The conditions may be imposed on sets with co-dimension larger than one or even
at just a single point. This appears to currently be the first such results in
a nonlocal setting with integrable kernels. The arguments used are direct, and
examples are provided demonstrating the explicit dependence of bounds for the
Poincar\'{e} constant upon structural parameters of the kernel and domain.
