  We introduce a general difference quotient representation for non-local
operators associated with a first-order linear operator. We establish new local
to non-local estimates and strong localization principles in various spaces of
functions, measures and distributions, which fully generalize those known for
gradients. Under suitable assumptions, we also establish the invariance of
quasiconvexity within the proposed local-nonlocal setting. Applications to the
fine properties of $\mathcal A$-gradient measures are further discussed.
