  A Galilean contraction is a way to construct Galilean conformal algebras from
a pair of infinite-dimensional conformal algebras, or equivalently, a method
for contracting tensor products of vertex algebras. Here, we present a
generalisation of the Galilean contraction prescription to allow for inputs of
any finite number of conformal algebras, resulting in new classes of
higher-order Galilean conformal algebras. We provide several detailed examples,
including infinite hierarchies of higher-order Galilean Virasoro algebras,
affine Kac-Moody algebras and the associated Sugawara constructions, and
$W_{3}$ algebras.
