Hydrogel colloids, i.e., micro- or nano-gels, are increasingly engineered as promising vehicles for polymer-based drug delivery systems. We report a continuum theory of deswelling dynamics of nanocomposite microgels driven by external osmotic shocks and further develop a universal framework, by introducing a buffer release domain, to quantitatively characterize a continuous drug release from deswollen microgels towards surroundings. The drug release is shown to proceed accompanied by an active outward solvent flow created by the elastically shrunken gel network. We further find that a declining trend in the cumulative release plateau with the drug size is followed by an apparent increase again as the drug size increases above a threshold. These findings highlight a nontrivial behavior that the resulting hydrodynamic interactions coexist collaboratively with the passive diffusions to facilitate a desired drug release. We show that deswelling of a stiffer microgel (the mesh size reduces slowly) or loading the larger drugs could bring a control-like release type, otherwise a burst-like release type emerges. Compared with a uniform microgel, the fuzzy-corona-like microgel enables a more productive drug release before reaching deswelling equilibrium. Our model not only predicts well the existing experiments, but also serves as a versatile paradigm to help understand the reciprocal roles of the solvent flow, the gel dynamics, and the diffusions in the polymer-based drug delivery systems.