The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is key for building scalable quantum information systems . In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation . We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state . Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits and a toric code state on a torus with sixteen data and eight ancillary qubits . Finally, we use this architecture to realize a hybrid analogue-digital evolution and use it for measuring entanglement entropy in quantum simulations , experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars . Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.