The operational lifetime of filtration membranes is reduced by the clogging of pores and subsequent build-up of a fouling or cake layer. Designing membrane operations in which clogging is delayed or even mitigated completely, requires in-depth insight into its origins. Due to the complexity of the clogging process, simplified model membranes fabricated in microfluidic chips have emerged as a powerful tool to study how clogs emerge and deteriorate membrane efficiency. However, to date, these have focussed solely on dead-end filtration, while cross-flow filtration is of greater practical relevance at the industrial scale. As such, the microscopic mechanisms of clogging in crossflow geometries have remained relatively ill-explored. Here we use a microfluidic filtration model to probe the kinetics and mechanisms of clogging in crossflow. Our study exposes two findings: (i) the primary clogging rate of individual pores depends only on the trans-membrane flux, whose strong effects are explained quantitatively by extending existing models with a term for flux-controlled flow-enhanced barrier crossing, (ii) cross-membrane flow affects the pore-pore communication, leading to a transition from correlated to uncorrelated clogging of the membrane, which we explain qualitatively by deriving a dimensionless number which captures two essential regimes of clogging at the microscale.