Is it permissible to pray in a moving bus, and is it obligatory to repeat the prayer if the worshipper arrives [at their destination] before the prayer's time expires?

The default ruling is that prayer is not permissible on a moving bus, due to the inability to fulfill its conditions and pillars completely, unless one fears missing the company (of fellow travelers) and the time (of prayer), in which case prayer is permissible, but it must be repeated. As for prayer on a ship, train, or airplane, it is valid if it is possible to perform the prayer perfectly.