We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.