  We demonstrate a SWAP gate between laser-cooled ions in a segmented microtrap
via fast physical swapping of the ion positions. This operation is used in
conjunction with qubit initialization, manipulation and readout, and with other
types of shuttling operations such as linear transport and crystal separation
and merging. Combining these operations, we perform quantum process tomography
of the SWAP gate, obtaining a mean process fidelity of 99.5(5)\%. The swap
operation is demonstrated with motional excitations below 0.05(1)~quanta for
all six collective modes of a two-ion crystal, for a process duration of
42~$\mu$s. Extending these techniques to three ions, we reverse the order of a
three-ion crystal and reconstruct the truth table for this operation, resulting
in a mean process fidelity of 99.96(13)\% in the logical basis.
