We present a versatile method for continuous-flow, on-chip biological processing of cells, large bio-particles, and functional beads. Using an asymmetric post array in pressure-driven microfluidic flow, we can move particles of interest across multiple, independent chemical streams, enabling sequential chemical operations. With this method, we demonstrate on-chip cell treatments such as labeling and washing, and bacterial lysis and chromosomal extraction. The washing capabilities of this method are particularly valuable because they allow many analytical or treatment procedures to be cascaded on a single device while still effectively isolating their reagents from cross-contamination.