We report measurements of velocity-dependent cross sections for vibrotationally inelastic scattering in the system Li-2 A(1) Sigma(u)(+)(v(i),j(i))) + X --> Li-2 A(1) Sigma(u)(+)(v(f),j(f)) + X, with v(i) = 9, j(i) = 42, and X = Xe, Ar, and Ne. The measurements range over a factor of 30 in energy. Quantum levels were chosen to elucidate the quasi-resonant vibrotational transfer process studied previously. A reduction in collision velocity results in both an increase in total vibrationally inelastic cross section and an enhancement of the quasi-resonant effect, with final rotational state distributions as narrow as 2.5HBAR (fwhm) observed. The largest cross section for Delta v < 0 is given by the formula Delta j(peak) = -4 Delta v and possesses a roughly 1/v(rel) velocity dependence. For Delta v = +1, energy thresholds shift the peak cross section at low velocity to the nearly energy resonant value of Delta j(peak) = -6. The similarity of the final state distributions for different target gases observed in previously measured rate constants does not hold in the velocity-dependent data; this results in part from the appearance of low-velocity dynamical thresholds for exoergic cross sections in the Ar and Ne systems. We compare the experimental results for X = Ne with cross sections calculated from quasi-classical trajectories on an ab initio potential energy surface; for j(f) greater than or equal to 46, agreement is quantitative, while for j(f) < 46, the calculation overestimates the cross sections.