Mediates the nuclear export of proteins (cargos), such as EIF5A, SMAD3 and isoform M2 of PKM (PKM2) (PubMed:10944119, PubMed:16449645, PubMed:26787900). In the nucleus binds cooperatively to its cargo and to the GTPase Ran in its active GTP-bound form. Docking of this trimeric complex to the nuclear pore complex (NPC) is mediated through binding to nucleoporins (PubMed:10944119, PubMed:16449645). Upon transit of a nuclear export complex into the cytoplasm, disassembling of the complex and hydrolysis of Ran-GTP to Ran-GDP (induced by RANBP1 and RANGAP1, respectively) cause release of the cargo from the export receptor (PubMed:10944119, PubMed:16449645). XPO4 then return to the nuclear compartment and mediate another round of transport (PubMed:10944119, PubMed:16449645). The directionality of nuclear export is thought to be conferred by an asymmetric distribution of the GTP- and GDP-bound forms of Ran between the cytoplasm and nucleus (PubMed:10944119, PubMed:16449645). Catalyzes the nuclear export of hypusinated EIF5A; a small cytoplasmic protein that enters nucleus and accumulates within nucleolus if not exported back by XPO4 (PubMed:10944119). Specifically mediates nuclear export of isoform M2 of PKM (PKM2) following PKM2 deacetylation by SIRT6 (PubMed:26787900). Also mediates the nuclear import of SOX transcription factors SRY and SOX2 (By similarity). {ECO:0000250|UniProtKB:Q9ESJ0, ECO:0000269|PubMed:10944119, ECO:0000269|PubMed:16449645, ECO:0000269|PubMed:26787900}. This is the function of ENSG00000132953 (XPO4, exportin 4).