This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial and boundary value problem associated to the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in $L^\infty$ and the advection vector field $v$ has low regularity properties, namely $v\in L^1(]0,T[,(W^{1,1}(\O))^d)$, with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in $L^p(\O)$ with $p<+\infty$, of the approximate solution towards the unique weak solution of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory.