In this Letter we provide a theory to show that high-field electrokinetic translocation of DNA through nanopores or nanochannels causes large transient variations of the ionic concentrations in front and at the back of the DNA due to concentration polarization (CP). The CP causes strong local conductivity variations, which can successfully explain the nontrivial current transients and ionic distributions observed in molecular dynamics simulations of nanopore DNA translocations as well as the transient current dips and spikes measured for translocating hairpin DNA. Most importantly, as the future of sequencing of DNA by nanopore translocation will be based on time-varying electrical conductance, CP, must be considered in experimental design and interpretation\char22{}currently these studies are mostly based on the incomplete pore conductance models that ignore CP and transients in the electrical conductance.