We study experimentally and numerically the dynamics of the spin ice material Dy$_2$Ti$_2$O$_7$ in the low temperature ($T$) and moderate magnetic field ($\mathbf{B}$) regime ($T\in[0.1,1.7]K$, $B\in[0,0.3]T$). Our objective is to understand the main physics shaping the out-of-equilibrium magnetisation vs. temperature curves in two different regimes... Very far from equilibrium, turning on the magnetic field after having cooled the system in zero field (ZFC) can increase the concentration of magnetic monopoles; this accelerates the dynamics. Similarly to electrolytes, this occurs through dissociation of bound monopole pairs. However, for spin ices the polarisation of the vacuum out of which the monopole pairs are created is a key factor shaping the magnetisation curves, with no analog. We observe a threshold field near $0.2T$ for this fast dynamics to take place, linked to the maximum magnetic force between the attracting pairs. Surprisingly, within a regime of low temperatures and moderate fields, an extended Ohm's law can be used to describe the ZFC magnetisation curve obtained with the dipolar spin-ice model. However, in real samples the acceleration of the dynamics appears even sharper than in simulations, possibly due to the presence of avalanches. On the other hand, the effect of the field nearer equilibrium can be just the opposite to that at very low temperatures. Single crystals, as noted before for powders, abandon equilibrium at a blocking temperature $T_B$ which increases with field. Curiously, this behaviour is present in numerical simulations even within the nearest-neighbours interactions model. Simulations and experiments show that the increasing trend in $T_B$ is stronger for $\mathbf{B}\parallel[100]$. This suggests that the field plays a part in the dynamical arrest through monopole suppression, which is quite manifest for this field orientation. (read more)