  For an oriented $4$--dimensional fiber bundle over $S^{1}$, we build a
relative trisection from a sutured Heegaard splitting of the fiber. We provide
an algorithm to explicitly construct the associated relative trisection
diagram, from a sutured Heegaard diagram of the fiber. As an application, we
glue our relative trisection diagrams with existing diagrams to recover
trisected closed fiber bundles over $S^1$ and trisected spun manifolds, and to
provide trisections for $4$--dimensional open-books.
