We solve the equations of trap-controlled tracer diffusion in amorphous solids in the case of constant empty-trap density. We show that the early-time tracer profiles develop exponential wings whose widths are given by the mean atomic displacement between trapping events. The wing amplitude increases linearly with time. In the long-time limit, the solutions are identical to the trap-free case, but with an effective diffusion coefficient that can be calculated from features of the early-time tracer profiles