Substrate-transferred crystalline coatings have emerged as a groundbreaking new concept in optical interference coatings. Building upon our initial demonstration of this technology in 2013, we have recently realized significant improvements in the optical performance of these novel single-crystal GaAs/AlGaAs multilayers. In the near-infrared, for center wavelengths spanning 1064 to 1560 nm, we have reduced the excess optical losses (scatter + absorption) to less than 5 ppm, with the direct measurement of sub-ppm optical absorption in these films, enabling the realization of a cavity finesse exceeding 600,000 at the telecom-relevant wavelength range near 1550 nm. In this presentation we outline preliminary measurements of the laser-induced damage threshold (LIDT) of these novel semiconductor-based interference coatings. For pulsed excitation (ns pulse durations at 1064 nm), the narrow bandgap of the constituent mirror materials limits the LIDT to 3-5 J/cm2. Under these conditions, laser damage is driven by two-photon absorption (TPA) in the semiconductor multilayer, primarily the high-refractive-index GaAs films. Note that improved performance may be realized for illumination wavelengths >1740 nm, where TPA is eliminated. For continuous-wave (CW) illumination, the high thermal conductivity (~30 Wm-1K-1) and low intrinsic absorption yield the potential for excellent performance. Here we present preliminary CW damage measurements for a 10-ppm transmission quarter-wave GaAs/AlGaAs Bragg mirror transferred to super-polished fused silica, with only a 1.4 K temperature rise for an intensity of ~1.5 MW/cm2. Further efforts will continue to push the limits of the structure with the aim of determining the maximum CW intensity that such mirrors can tolerate.