We investigate uniform versions of metric regularity and strong metric regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a parametric generalized equation or, more generally, of a differential generalized equation (DGE). The latter model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We propose two inexact path-following methods for DGEs having the order of the grid error $O(h)$ and $O(h^2)$, respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics.
Further, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory by focusing on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts.