  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.
