  We prove some $\ell$-independence results on local constancy of \'etale
cohomology of rigid analytic varieties. As a result, we show that a closed
subscheme of a proper scheme over an algebraically closed complete
non-archimedean field has a small open neighborhood in the analytic topology
such that, for every prime number $\ell$ different from the residue
characteristic, the closed subscheme and the open neighborhood have the same
\'etale cohomology with $\mathbb Z/\ell \mathbb Z$-coefficients. The existence
of such an open neighborhood for each $\ell$ was proved by Huber. A key
ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the
compatibility of the nearby cycles over general bases with base change.
