Quantum information processing will require combinations of gate operations and communication, with each applied in parallel to large numbers of quantum systems. These tasks are often performed sequentially, with gates implemented by pulsed fields and information transported either by moving the physical qubits or using photonic links. For trapped ions, an alternative approach is to implement quantum logic gates by transporting the ions through static laser beams, combining qubit operations with transport. This has significant advantages for scalability since the voltage waveforms required for transport can potentially be generated using micro-electronics integrated into the trap structure itself, while both optical and microwave control elements are significantly more bulky. Using a multi-zone ion trap, we demonstrate transport gates on a qubit encoded in the hyperfine structure of a beryllium ion. We show the ability to perform sequences of operations, and to perform parallel gates on two ions transported through separate trap locations using a single recycled laser beam. For the latter, we demonstrate independent quantum gates by controlling the speed of each of the ions. This work provides a scalable path to ion trap quantum computing without a dramatic increase in optical control complexity.