We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with 'unstructured' overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.