Pump to separate protein/DNA instead of electrophoresis?
If you tried to set up a system in which movement of DNA molecules through a gel matrix was driven by pumping buffer, the gel would become compressed. I am old enough to remember running vertical agarose gel electrophoresis overnight, and these gels would collapse under normal pressure to about 2/3rds of their initial height. A gel porous enough to give useful separation of DNA molecules just isn't strong enough to withstand pumping.