Carrier dynamics in single quasi-1D GaAs quantum wires are studied in a wide temperature range by near-field scanning optical microscopy using pico- and femtosecond pulses. Luminescence and pump-probe experiments with a spatial resolution of 250 nm and a time resolution of up to 200 fs allow for a separation of carrier transport along the quantum wire and in the embedding GaAs quantum well from local carrier relaxation phenomena. We demonstrate that local potential barriers close to the quantum wire strongly affect the lateral carrier transport into the quantum wire. This drift-diffusive motion occurs in the pico- to nanosecond regime with diffusion lengths of up to several microns. Diffusive transport along the quantum wire is characterized by carrier motion on a somewhat shorter picosecond time scale. In contrast, sub-picosecond relaxation times are found for the redistribution of carriers from high-lying to low-lying quantum wire states. This relaxation is governed by electron-electron and electron-phonon scattering.