  Let $\mathcal V$ be a discrete valuation ring of mixed characteristic with
perfect residue field. Let $X$ be a geometrically connected smooth proper curve
over $\mathcal V$. We introduce the notion of constructible convergent
$\nabla$-module on the analytification $X_{K}^{\mathrm{an}}$ of the generic
fibre of $X$. A constructible module is an $\mathcal
O_{X_{K}^{\mathrm{an}}}$-module which is not necessarily coherent, but becomes
coherent on a stratification by locally closed subsets of the special fiber
$X_{k}$ of $X$. The notions of connection, of (over-) convergence and of
Frobenius structure carry over to this situation. We describe a specialization
functor from the category of constructible convergent $\nabla$-modules to the
category of $\mathcal D^\dagger_{\hat X \mathbf Q}$-modules. We show that if
$X$ is endowed with a lifting of the absolute Frobenius of $X$, then
specialization induces an equivalence between constructible
$F$-$\nabla$-modules and perverse holonomic $F$-$\mathcal D^\dagger_{\hat X
\mathbf Q}$-modules.
