  We prove the rigidity of rectifiable boundaries with constant distributional
mean curvature in the Brendle class of warped product manifolds (which includes
important models in General Relativity, like the deSitter--Schwarzschild and
Reissner--Nordstrom manifolds). As a corollary we characterize limits of
rectifiable boundaries whose mean curvatures converge, as distributions, to a
constant. The latter result is new, and requires the full strength of
distributional CMC-rigidity, even when one considers smooth boundaries whose
mean curvature oscillations vanish in arbitrarily strong $C^k$-norms.
