Polymer flows through pores, nozzles and other small channels govern engineered and naturally occurring dynamics in many processes, from 3D printing to oil recovery in the earth's subsurface to a wide variety of biological flows. The crosslinking of polymers can change their material properties dramatically, and it is advantageous to know a priori whether or not crosslinking polymers will lead to clogged channels or cessation of flow. In this study, we investigate the flow of a common biopolymer, alginate, while it undergoes crosslinking by the addition of a crosslinker, calcium, driven through a microfluidic channel at constant flow rate. We map the boundaries defining complete clogging and flow as a function of flow rate, polymer concentration, and crosslinker concentration. Interestingly, the boundaries of the dynamic behavior qualitatively match the thermodynamic jamming phase diagram of attractive colloidal particles. That is, polymer clogging occurs in a region analogous to colloids in a jammed state, while the polymer flows in regions corresponding to colloids in a liquid phase. However, between the dynamic regimes of complete clogging and unrestricted flow, we observe a remarkable phenomenon in which the crosslinked polymer intermittently clogs the channel. This pattern of deposition and removal of a crosslinked gel is simultaneously highly reproducible, long-lasting, and controllable by system parameters. Higher concentrations of polymer and cross-linker result in more frequent ablation, while gels formed at lower component concentrations ablate less frequently. Upon ablation, the eluted gel maintains its shape, resulting in micro-rods several hundred microns long. Our results suggest both rich dynamics of intermittent flows in crosslinking polymers and the ability to control them.