The kinetics of colloidal gels under narrow confinement are of widespread practical relevance, with applications ranging from flow in biological systems to 3D printing. Although the properties of such gels under uniform shear have received considerable attention, the effects of strongly nonuniform shear are far less understood. Motivated by the possibilities offered by recent advances in nano- and microfluidics, we explore the generic phase behavior and dynamics of attractive colloids subject to microchannel flow, using mesoscale particle-based hydrodynamic simulations. Whereas moderate shear strengths result in shear-assisted crystallization, high shear strengths overwhelm the attractions and lead to melting of the clusters. Within the transition region between these two regimes, we discover remarkable dynamics of the colloidal aggregates. Shear-induced surface melting of the aggregates, in conjunction with the Plateau-Rayleigh instability and size-dependent cluster velocities, leads to a cyclic process in which elongated threads of colloidal aggregates break up and reform, resulting in large crystallites. These insights offer new possibilities for the control of colloidal dynamics and aggregation under confinement.