We consider the planar parabolic, hyperbolic, and highly-elliptic restricted three-body problems (R3BP). In particular, we formulate sufficient conditions for no transit (escape of the infinitesimal body) to occur in the near-parabolic R3BP using the disturbing function derived in Mamedov (1987) as our starting point. We take a middle approach between purely numerical and strictly analytic in the form of infinite series expansions to derive approximation mappings modeling the dynamics. Motivated by the long-term dynamics of the highly-elliptic R3BP, we study an annulus map with properties which are unexpectedly different from those of the standard annulus map.