Muon accelerators have a potential to open new paths for advancing particle physics both in intensity and energy frontier. Short lifetime of muons requires the implementation of ionization cooling. Most contemporary designs of muon ionization cooling channels require high-performing normal-conducting RF cavities in external multi-tesla solenoidal magnetic fields. One of the limitations of evacuated cavities' performance is RF breakdown. It was experimentally shown that the presence of external magnetic field increases breakdown rate at a given gradient. Pulsed heating model of RF breakdown aims to explain the deterioration of cavity gradients in strong focusing magnetic fields. The model proposes that cyclic fatigue leads to the metal surface degradation and eventual breakdown. The cyclic fatigue is caused by the periodic bombardment of inner surfaces of the cavity by dark current electrons, emitted from local surface imperfections and focused by external magnetic field.,An 805~MHz pillbox cavity was designed to study the effect of external magnetic field on RF breakdown. Special "modular" design of the cavity allowed for control over sources of systematic error and frequent turnaround times for surface inspections. Frequent inner surface inspections are essential for the breakdown damage tracking and understanding the effect of different materials on RF breakdown. In this work I will discuss the results of high power tests of the Modular Cavity with copper and beryllium walls in the range of magnetic fields. Measured gradient performance is tested against predictions of pulsed heating model, modified for experimental conditions of the Modular Cavity.