  We develop techniques to capture the effect of transport on the long-term
dynamics of small, localized initial data in nonlinearly coupled
reaction-diffusion-advection equations on the real line. It is well-known that
quadratic or cubic nonlinearities in such systems can lead to growth of small,
localized initial data and even finite time blow-up. We show that, if the
components exhibit different velocities, then quadratic or cubic mix-terms,
i.e. terms with nontrivial contributions from both components, are harmless. We
establish global existence and diffusive Gaussian-like decay for exponentially
and algebraically localized initial conditions allowing for quadratic and cubic
mix-terms. Our proof relies on a nonlinear iteration scheme that employs
pointwise estimates. The situation becomes very delicate if other quadratic or
cubic terms are present in the system. We provide an example where a quadratic
mix-term and a Burgers'-type coupling can compensate for a cubic term due to
differences in velocities.
