  Given a central isogeny $\pi\colon G\to H$ of connected reductive
$\overline{\mathbb Q}_p$-groups, and a local Galois representation $\rho$
valued in $H(\overline{\mathbb Q}_p)$ that is trianguline in the sense of
Daruvar, we study whether a lift of $\rho$ along $\pi$ is still trianguline. We
give a positive answer under weak conditions on the Hodge--Tate--Sen weights of
$\rho$, and the assumption that the trianguline parameter of $\rho$ can be
lifted along $\pi$. This is an analogue of the results proved by Wintenberger,
Conrad, Patrikis, and Hoang Duc for $p$-adic Hodge-theoretic properties of
$\rho$. We describe a Tannakian framework for all such lifting problems, and we
reinterpret the existence of a lift with prescribed local properties in terms
of the simple connectedness of a certain pro-semisimple group. While applying
this formalism to the case of trianguline representations, we extend a result
of Berger and Di Matteo on triangulable tensor products of $B$-pairs.
