A space–time adaptive method is presented for the numerical simulation of mass transport in electroosmotic and pressure-driven microflows in two space dimensions. The method uses finite elements with large aspect ratio, which allows the electroosmotic flow and the mass transport to be solved accurately despite the presence of strong boundary layers. The unknowns are the external electric potential, the electrical double layer potential, the velocity field and the sample concentration. Continuous piecewise linear stabilized finite elements with large aspect ratio and the Crank–Nicolson scheme are used for the space and time discretization of the concentration equation. Numerical results are presented showing the efficiency of this approach, first in a straight channel, then in crossing and multiple T-form configuration channels.