The propagation of intense laser pulses with durations longer than the plasma period through tapered plasma channels is investigated theoretically and numerically. General propagation equations are presented and reduced partial differential equations that separately describe the forward Raman (FR) and self-modulation (SM) instabilities in a nonuniform plasma are derived. Local dispersion relations for FR and SM instabilities are used to analyze the detuning process arising from a longitudinal density gradient. Full-scale numerical fluid simulations indicate parameters that favorably excite either the FR or SM instability. The suppression of the FR instability and the enhancement of the SM instability in a tapered channel in which the density increases longitudinally is demonstrated. For a pulse undergoing a self-modulation instability, calculations show that the phase velocity of the wakefield in an untapered channel can be significantly slower than the pulse group velocity. Simulations indicate that this wake slippage can be forestalled through the use of a tapered channel.