Dilute fixed fiber beds provide a model system for studying drop dynamics in disordered flows. Fluctuations about the mean uniform velocity are generated by fiber elements within the media, and the disturbance velocities far from any single fiber (at distances on the order of the pore size) have been predicted to be strong in terms of drop deformation and breakup by Mosler and Shaqfeh [Phys. Fluids 9, 5 (1997)]. In this work, we focus on the importance of near-field interactions, or the flow close to individual fibers. We present experimental observations of drop deformation and breakup during flow through a dilute bed of randomly placed fibers. We found breakup to result from only close interactions with fibers and describe two near-field breakup mechanisms which we term "graze" and "hairpin" processes. In addition, we present the breakup probability through the experimental fiber bed as a function of the appropriate Capillary number Ca. To better understand the near-field interactions, we used the boundary integral method to determine drop shape evolution in the flow around an infinite fiber within a porous medium, and our simulations capture the breakup mechanisms observed during experiments. To compare with experimental breakup probabilities, we have defined a critical offset for breakup during flow past a fiber and assuming straight center-of-mass trajectories, calculated breakup probabilities based on this simple model. These predictions compare well with the experimental measurements for Ca≥2.