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 Gal(Q p /Qp) with fixed Hodge-Tate weights and large L-invariant. Second, in dimension d, let Vn 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, σ) of V (or equivalently for any chosen triangulation of its attached (ϕ, Γ)-module Drig(V ) over the Robba ring), there exists a sequence of refinement σn of each of the Vn such that the limit as refined representations (Vn, σn) converges to the (V, σ). This result does not hold under the weaker assumption that Vn converges only uniformly p-adically to V (in the sense of Chenevier, Khare and Larsen).