We investigate mixtures of ring and linear polymers in solution at various number ratios, ranging from pure chains to pure rings, and at densities around the overlap concentration. In bulk and at rest, we find a shrinking of both topologies with increasing polymer content of the solution. At the same time, we observe an increase in the solution viscosity with a concomitant reduction of the polymer diffusivity. When exposing ring-chain mixtures of any composition to a pressure-driven flow in a slit channel, we find that ring polymers always migrate toward the center, whereas chains populate the regions of high local shear close to the channel walls. Interestingly, in a pure chain solution, this cross-stream migration toward the walls is absent. This phenomenon could be used to iteratively separate chains and rings at high mass throughput in simple microfluidic devices using pressure-driven flows. Furthermore, we show how pressure-driven flow can be used as a computationally efficient approach for determining the shear viscosity of (complex) liquids.