We report an electrical transport and electroluminescence (EL) spectroscopy study of single-barrier GaAs-AlAs-GaAs $p\ensuremath{-}i\ensuremath{-}n$ tunnel structures with a barrier thickness in the range 4.5--8.0 nm. The results permit us to determine directly the relative roles of nonresonant \ensuremath{\Gamma}-\ensuremath{\Gamma} tunneling and resonant $\ensuremath{\Gamma}\ensuremath{-}X\ensuremath{-}\ensuremath{\Gamma}$ intervalley transfer in the transport through the indirect-gap tunnel barriers. The $\ensuremath{\Gamma}\ensuremath{-}X\ensuremath{-}\ensuremath{\Gamma}$ transport is shown to take place predominantly without conservation of transverse wave vector ${(k}_{\ensuremath{\parallel}}),$ with ${k}_{\ensuremath{\parallel}}$-conserving scattering via ${X}_{z}$ states only significantly close to the onset of $\ensuremath{\Gamma}\ensuremath{-}X$ intervalley transfer. By detecting extremely weak EL arising from excited ${X}_{z}$ states we show that the complete $\ensuremath{\Gamma}\ensuremath{-}X\ensuremath{-}\ensuremath{\Gamma}$ transport process is very strongly sequential and determine, quantitatively, the comparative time scales for $X\ensuremath{-}\ensuremath{\Gamma}$ and inter-X-level scattering. The bias-dependent $\ensuremath{\Gamma}\ensuremath{-}X\ensuremath{-}\ensuremath{\Gamma}$ and \ensuremath{\Gamma}-\ensuremath{\Gamma} transport times are determined for AlAs barrier widths in the range 3.0--10 nm. The intervalley $\ensuremath{\Gamma}\ensuremath{-}X\ensuremath{-}\ensuremath{\Gamma}$ transport model yields results in good agreement with experiment and demonstrates that, providing intervalley $\ensuremath{\Gamma}\ensuremath{-}X$ transfer is energetically possible, nonresonant \ensuremath{\Gamma}-\ensuremath{\Gamma} tunneling only contributes significantly to the transport characteristics for barrier widths of \ensuremath{\sim}3 nm or less.