  We use Scholze's framework of diamonds to gain new insights in
correspondences between $p$-adic vector bundles and local systems. Such
correspondences arise in the context of $p$-adic Simpson theory in the case of
vanishing Higgs fields. In the present paper we provide a detailed analysis of
local systems on diamonds for the \'etale, pro-\'etale, and the $v$-topology,
and study the structure sheaves for all three topologies in question. Applied
to proper adic spaces of finite type over $\mathbb{C}_p$ this enables us to
prove a category equivalence between $\mathbb{C}_p$-local systems with integral
models, and modules under the $v$-structure sheaf which modulo each $p^n$ can
be trivialized on a proper cover. The flexibility of the $v$-topology together
with a descent result on integral models of local systems allows us to prove
that the trivializability condition in the module category may be checked on
any normal proper cover. This result leads to an extension of the parallel
transport theory by Deninger and the second author to vector bundles with
numerically flat reduction on a proper normal cover.
