  Recently, a class of non-Lorentzian supersymmetric Lagrangian field theories
was considered, in some cases describing M-theory brane configurations, but
more generally found as fixed points of non-Lorentzian RG flows induced upon
Lorentzian theories. In this paper, we demonstrate how the dynamics of such
theories can be reduced to motion on the supersymmetric moduli space of BPS
solitons of the parent theory. We focus first on the $\mathcal{N}=(1,1)$
$\sigma$-model in $(1+1)$-dimensions with potential, where we produce a
supersymmetric extension to the standard geodesic approximation for slow kink
motion. We then revisit the $(4+1)$-dimensional Yang-Mills-like theory with 24
supercharges describing a null compactification of M5-branes. We show that the
theory reduces to a $\sigma$-model on instanton moduli space, extended by
couplings to additional fields from the parent theory, and possessing (8+8)
super(conformal) symmetries along with 8 further fermionic shift symmetries. We
derive this model explicitly for the single $SU(2)$ instanton.
