A technique is presented that is ideally suited for characterizing the mechanical and transport properties of polymer gels at small strains. A flat, circular punch and a flat, rectangular punch are used to probe the response of gels under oscillatory loading conditions. Solvent transport within the gel is driven by gradients in hydrostatic pressure, giving rise to a dissipative response quantified by the phase lag between the punch displacement and the resulting load. By comparing results for different punch sizes, it is possible to differentiate between dissipation resulting from internal solvent flow and dissipation due to the viscoelastic character of the polymer network itself. Use of the technique is illustrated with poly(n-isopropylacrylamide) gels, which undergo a reversible structural transition just above room temperature. We show that heterogeneous structure formed above the transition temperature is not conducive to internal solvent flow within these gels.