The actin and microtubule cytoskeletons form active networks in the cell that can contract and remodel, resulting in vital cellular processes such as cell division and motility. Motor proteins play an important role in generating the forces required for these processes, but more recently the concept of passive cross-linkers being able to generate forces has emerged. So far, these passive cross-linkers have been studied in the context of separate actin and microtubule systems. Here, we show that cross-linkers also allow actin and microtubules to exert forces on each other. More specifically, we study single actin filaments that are cross-linked to growing microtubule ends, using in vitro reconstitution, computer simulations, and a minimal theoretical model. We show that microtubules can transport actin filaments over large (micrometer-range) distances and find that this transport results from two antagonistic forces arising from the binding of cross-linkers to the overlap between the actin and microtubule filaments. The cross-linkers attempt to maximize the overlap between the actin and the tip of the growing microtubules, creating an affinity-driven forward condensation force, and simultaneously create a competing friction force along the microtubule lattice. We predict and verify experimentally how the average transport time depends on the actin filament length and the microtubule growth velocity, confirming the competition between a forward condensation force and a backward friction force. In addition, we theoretically predict and experimentally verify that the condensation force is of the order of 0.1 pN. Thus, our results reveal an active mechanism for local actin remodeling by growing microtubules that relies on passive cross-linkers.