Material contrasts across faults are a common occurrence, and it is important to understand if these material contrasts can influence the path of rupture propagation. Here we examine models, solved numerically, of rupture propagation through one type of geometric complexity, that of a fault branch stemming from a planar main fault on which rupture initiates. This geometry, with a material contrast across the main fault, could be representative of either a mature strike-slip fault or a subduction zone interface. We consider branches in both the compressional and extensional quadrants of the fault, and material configurations in which the branch fault is in either the stiffer or the more compliant material as well as configurations with no material contrast. We find that there are regimes in which this elastic contrast can influence the rupture behavior at a branching junction, but there are also stress states for which the branch activation will not depend on the orientation of the mismatch. For the scenarios presented here, both compressional and extensional side branches are more likely to rupture if the branch is on the side of the fault with the more compliant material versus the stiffer material. The stresses induced on the branch fault, by rupture traveling on the main fault, are different for the two orientations of material contrast. We show how the interactions between rupture on the two faults determine which faults are activated.