We report on the nonequilibrium monopole dynamics in the classical spin ice Dy₂Ti₂O₇ detected by means of high-resolution magnetostriction measurements. Significant lattice changes occur at the transition from the kagome-ice to the saturated-ice phase, visible in the longitudinal and transverse magnetostriction. A hysteresis opening at temperatures below 0.6K suggests a first-order transition between the kagome and saturated state. Extremely slow lattice relaxations, triggered by changes of the magnetic field, were observed. These lattice-relaxation effects result from nonequilibrium monopole formation or annihilation processes. The relaxation times extracted from our experiment are in good agreement with theoretical predictions with decay constants of the order of 104 s at 0.3K.