We have studied the field-dependent ac magnetic susceptibility of single crystals of Dy2Ti2O7 spin ice along the [111] direction in the temperature range 1.8-7 K. Our data reflect the onset of local spin-ice order in the appearance of different field regimes. In particular, we observe a prominent feature at approximately 1.0 T that is a precursor of the low-temperature metamagnetic transition out of field-induced kagome ice, below which the kinetic constraints imposed by the ice rules manifest themselves in a substantial frequency dependence of the susceptibility. Despite the relatively high temperatures, our results are consistent with a monopole picture, and they demonstrate that such a picture can give physical insight into spin-ice systems even outside the low-temperature, low-density limit where monopole excitations are well-defined quasiparticles.