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In this article, we prove the existence of rigid analytic families of$G$-stable lattices with locally constant reductions inside families ofrepresentations of a topologically compact group $G$, extending a result ofHellman obtained in the semi-simple residual case. Implementing thisgeneralization in the context of Galois representations, we prove a localconstancy result for reductions modulo prime powers of triangulinerepresentations of generic dimension $d$. Moreover, we present two explicitapplications. First, in dimension two, we extend to a prime power setting andto the whole rigid projective line a recent result of Bergdall, Levin and Liuconcerning reductions of semi-stable representations of$\text{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ with fixed Hodge-Tateweights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$be a sequence of crystalline representations converging in a certain geometricsense to a crystalline representation $V$. We show that for any refined version$(V, \sigma)$ of $V$ (or equivalently for any chosen triangulation of itsattached $(\varphi, \Gamma)$-module $D_{\text{rig}} (V)$ over the Robba ring),there exists a sequence of refinement $\sigma_n$ of each of the $V_n$ such thatthe limit as refined representations $(V_n , \sigma_n )$ converges to the $(V,\sigma)$. This result does not hold under the weaker assumption that $V_n$converges only uniformly $p$-adically to $V$ (in the sense of Chenevier, Khareand Larsen).

On the existence of analytic families of stable lattices in trianguline representations and their reductions