  In this article, we prove the existence of rigid analytic families of
$G$-stable lattices with locally constant reductions inside families of
representations of a topologically compact group $G$, extending a result of
Hellman obtained in the semi-simple residual case. Implementing this
generalization in the context of Galois representations, we prove a local
constancy result for reductions modulo prime powers of trianguline
representations of generic dimension $d$. Moreover, we present two explicit
applications. First, in dimension two, we extend to a prime power setting and
to the whole rigid projective line a recent result of Bergdall, Levin and Liu
concerning reductions of semi-stable representations of
$\text{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ with fixed Hodge-Tate
weights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$
be a sequence of crystalline representations converging in a certain geometric
sense to a crystalline representation $V$. We show that for any refined version
$(V, \sigma)$ of $V$ (or equivalently for any chosen triangulation of its
attached $(\varphi, \Gamma)$-module $D_{\text{rig}} (V)$ over the Robba ring),
there exists a sequence of refinement $\sigma_n$ of each of the $V_n$ such that
the 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, Khare
and Larsen).
