We study the long-time behavior for continuous-time Markov processes on the state space R≥0: = [0;∞), which arise as unique strong solutions to stochastic equations with jumps. We establish, under a global dissipativity condition combined with a comparison principle, exponential ergodicity in various Wasserstein distances on R≥0. Our main emphasis lies on the derivation of these estimates under minimal moment conditions to be imposed on the associated Lévy measures of the noises. We apply our method to continuous-state branching processes with immigration (shorted as CBI processes), to nonlinear CBI processes, and finally to CBI processes in Lévy random environments.