The fate of Many-Body Localization (MBL) in the thermodynamic limit remains elusive, partly because numerical studies suffer from unexplained finite-size effects. We introduce and numerically study the (CTC)—a quantity that upper bounds the number of particles that can ever be transported across a central cut of a one-dimensional lattice. For ergodic systems of spinless fermions, the CTC is linear in the system size $L$, while we expect it to be $O(1)$ for localized models. Surprisingly, in the interacting Anderson model for numerically accessible $L$, the disorder-averaged CTC is small, but grows with $L$ at an increasing rate. Moreover, this growth rate appears to be independent of the disorder strength $W$ at very large $W$. We find that, for these system sizes, this growth occurs because, as $L$ increases, many-body resonances that transport more charge across the cut become more likely. Using a diagonally-improved perturbative model for the weakly interacting regime, we provide an understanding of the microscopic origins of the growth of these (CTRs). We find that these resonances are sensitive to charge configurations over a spatial region whose size is set by the range of the resonance, not by $W$, and that numerics cannot access system sizes where their behavior will converge. However, this effective model is consistent with a regime of strong disorder where, for large $L$, resonances are exponentially suppressed in their size. Finally, we study measures of average charge transport and suggest that for strong enough disorder, average product states can only transfer $O(1)$ charge. Our work suggests that the unsettled growth of short-ranged many-body resonances with $L$ contributes to the finite-sized drift towards thermalization at numerically accessible system sizes, and provides an understanding of how they can remain controlled or eventually destabilize the MBL phase.