  We show that infinite families of non-relativistic spin-$3$ symmetries in
$2+1$ dimensions, which include higher-spin extensions of the Bargmann,
Newton-Hooke, non-relativistic Maxwell, and non-relativistic AdS-Lorentz
algebras, can be obtained as Lie algebra expansions of two different spin-$3$
extensions of the Nappi-Witten symmetry. These higher-spin Nappi-Witten
algebras, in turn, are obtained by means of In\"on\"u-Wigner contractions
applied to suitable direct product extensions of $\mathfrak{sl}(3,\mathbb{R})$.
Conversely, we show that the same result can be obtained by considering
contractions of expanded $\mathfrak{sl}(3,\mathbb{R})$ algebras. The method can
be used to define non-relativistic higher-spin Chern-Simon gravity theories in
$2+1$ dimensions in a systematic way.
