This paper studies a class of nonlinear variable-order fractional boundary value problems involving the Riemann–Liouville fractional derivative. An infinite-partition approximation is introduced by replacing the variable-order function with a sequence of piecewise constant functions, transforming the original problem into a family of constant-order fractional boundary value problems. For each approximating problem, an equivalent integral equation and the corresponding Green function are derived. Existence, uniqueness, and Ulam–Hyers stability of solutions are established using Schauder's fixed-point theorem and the Banach contraction principle. It is further proved that the approximating solutions converge uniformly to a solution of the original variable-order problem, providing a rigorous connection between constant-order and variable-order fractional models. A numerical example illustrates the theoretical results.