  We study Thouless pumping and arresting of gap solitons in a two-component
Bose gas loaded into an optical superlattice. We show that, depending on the
atomic interactions and chemical potentials, the two solitons can be
simultaneously pumped or trapped, but we also identify regimes where one
soliton is pumped and the other arrested. These behaviors can be understood by
considering an effective model of the system based on a variational approach,
which reveals that the soliton width is a suitable qualifier for the observed
behaviours: solitons with a larger width get transported, while solitons with a
smaller width get trapped. Since these width can be controlled by tuning
interactions, it should be possible to observe all behaviours experimentally.
