Li₂ , HF, and H₂ and other light diatomics at high vibration-rotational excitation exhibit an unusual form of vibration-rotation transfer generally associated with near energy resonance. This quasi-resonant transfer (QRT) gives rise to narrow rotational distributions. Near resonance in angular momentum is also a necessary requirement for the occurrence of QRT. However, the underlying physical processes differ little from those governing the more common forms of collisional transfer which, along with QRT, can be rationalized via the mechanism of linear to angular momentum interconversion within boundary conditions set by energy conservation. Velocity- j plots illustrate that these boundary conditions moderate the mechanism in a unique fashion in the case of QRT since they are sharply defined around a limited set of j values. The occurrence of QRT will be widespread in the high lying states of light diatomic molecules, the hydrides for example, and may readily be identified using plots of the energy and angular momentum conservation relations. Energy conservation forces a reduction in the maximum available torque arm in the angular momentum mechanism for all but a narrow range of j transitions. This analysis of the primary physical mechanism is confirmed via multiellipsoid Monte Carlo calculations for Li₂ and for H₂ . In HF-Ar we show that QRT is a much more likely process than pure rotational transfer giving rise to collisional pumping which will be enhanced in a multicollision environment.