We explore charge-trapping effects in cryogenic particle detectors composed of single-crystal silicon substrates with both titanium transition-edge sensors (TES) and charge-collection electrodes deposited upon them. These effects include transients on various time scales which follow the evolution of different kinds of space charge, intrinsic gain and linearity shifts in signals characteristic of changes in the absorption of energy carried by electrons and holes, variations in charge-collection efficiency and ionization resolution, etc., The physics involved, relevant for many other cryogenic, semiconductor-based devices, includes a variety of charge trapping and transport mechanisms.