  We study geodesics in the complete family of expanding impulsive
gravitational waves propagating in spaces of constant curvature, that is
Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous
form of the metric we rigorously prove existence and global uniqueness of
continuously differentiable geodesics (in the sense of Filippov) and study
their interaction with the impulsive wave. Thereby we justify the
"$C^1$-matching procedure" used in the literature to derive their explicit
form.
