The user should be able to set whether they want pages/tabs to be moved in the background, without their view switching to the tab group that the page/tab was moved to. This would smoothen workflows and prevent users having to manually click or by shortcut switch back to the other window/tab group. Screen recording below for reference.
This could be toggled on/off on either the Tabs or Browsing settings panes.
Tabs
Browsing
