For dynamic screen changes in an SPA, I highly recommend setting focus to the region with new content, preferably to a top-level heading within the new content. An ARIA live region will read new content to a screen reader user but the keyboard focus will remain in a illogical location on the screen, and very often focus will be lost entirely since content is removed. Reading out all the new content may also be overwhelming for the user.
Also, when a screen changes in an SPA, the page title should be updated. A screen reader user will not be notified of this change which is why focusing a new top level heading for new content is important.