  We investigate the consistency and convergence of flux-corrected finite
element approximations in the context of nonlinear hyperbolic conservation
laws. In particular, we focus on a monolithic convex limiting approach and
prove a Lax--Wendroff-type theorem for the corresponding semi-discrete problem.
A key component of our analysis is the use of a weak estimate on bounded
variation, which follows from the semi-discrete entropy stability property of
the method under investigation. For the Euler equations of gas dynamics, we
prove the weak convergence of the flux-corrected finite element scheme to a
dissipative weak solution. If a strong solution exists, the sequence of
numerical approximations converges strongly to the strong solution.
