The variation of hydrocarbon abundances with altitude, latitude, and season in Saturn's stratosphere is investigated using a one-dimensional, time-variable, photochemical model. The results indicate that hydrocarbon abundances at pressures less than ~0.01 mbar are extremely sensitive to solar flux variations due to changes in season, latitude, orbital radius, solar cycle, and ring shadowing. Long vertical diffusion times in the 0.01--1 mbar region introduce phase lags in the response to insolation changes. At pressures greater than 1 mbar, vertical diffusion timescales are longer than a Saturnian year. Therefore, relatively long-lived hydrocarbons like C2H2, C2H6, and C3H8 exhibit little seasonal variation in the lower stratosphere, and the yearly averaged solar insolation combined with vertical diffusion control species abundances in this region. Species with short photochemical lifetimes (e.g., CH3, C2H4) continue to experience seasonal variations, even at low altitudes. The model results are compared with infrared observations from the TEXES spectrometer at NASA's Infrared Telescope Facility. We find that the observed C2H2 mixing-ratio variation with latitude is reasonably well predicted, whereas the C2H6 distribution is poorly predicted by the models. Meridional transport is likely affecting the distribution of C2H6, which has a long photochemical lifetime, whereas the distribution of shorter-lived C2H2 is controlled by photochemistry and vertical diffusion. Given the different observed behavior of C2H2 and C2H6, we constrain meridional transport timescales at 2 mbar on Saturn to be in the range ~100--700 years, corresponding to meridional wind or diffusion speeds of ~0.4--2 cm s-1.