We show that infinite families of Galilean spin-3 symmetries in 2+1 dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, nonrelativistic Maxwell, and nonrelativistic 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 Inönü-Wigner contractions applied to suitable direct product extensions of sl(3,R). Conversely, we show that the same result can be obtained by considering contractions of expanded sl(3,R) algebras. The method can be used to define nonrelativistic higher-spin Chern-Simon gravity theories in 2+1 dimensions in a systematic way.