Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in an earlier published version of this paper meet such objectives for a class of parametrized problems. An extended version of the abstract mathematical framework is established rigorously here, together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal models of diffusion and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical discretizations to problems involving nonlocal interactions on multiple scales. This work provides new insight in this regard through a careful analysis of related conforming finite element discretizations, and the finding is valid under minimal regularity assumptions on exact solutions. It reveals that for the nonlocal models under consideration and their local limit, as long as the finite element space contains continuous piecewise linear functions, the Galerkin finite element approximation is always asymptotically compatible. For piecewise constant finite elements, whenever applicable, it is shown that a correct local limit solution can also be obtained as long as the discretization (mesh) parameter decreases faster than the modeling (horizon) parameter does. These results can be used to guide future computational studies of nonlocal problems. Some other applications, such as the fractional PDE limit of nonlocal models, and open questions are also presented