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The lead-free halide double perovskite Cs₂AgBiBr₆ has emerged as a promising candidate for applications in tandem perovskite solar cells. The measured low carrier mobility, under 11 cm²/Vs, poses a challenge in developing efficient devices. Furthermore, the relative importance of defects and phonons in the scattering of the charge carriers remains unclear. In this first-principles investigation, we employ density functional perturbation theory (DFPT), Wannier-Fourier interpolation, and the Boltzmann transport equation (BTE) to calculate the electronic and phonon band structures, electron-phonon vertices, and intrinsic mobilities of electrons and holes in Cs₂AgBiX₆ (X=Br,Cl) at finite temperature. We find that phonon scattering accounts for the measured mobility at room temperature, and we identify the dominant electron-phonon scattering process in Cs₂AgBiX₆ (X=Br,Cl). Our findings provide an atomic-scale explanation for the low intrinsic carrier mobilities in these important solar cell candidate materials.