Answer
This is controlled by the Custom Action Configuration behind that button's event/action — specifically its 'LU Dependencies' setting, where you specify which Logical Units (entities) should be treated as needing a refresh whenever that action runs. Add the relevant LU(s) as dependencies there, and the connected Entitysets on the page will automatically refresh after the action executes, without needing a full browser reload. Note that only entities actually related to the page's underlying projection will be available to select as dependencies.