It is usually preferable to capture LOC first, then capture the GS from below.
However, if the GS bar is centered whilst descending on to the ILS, it is perfectly normal to adjust vertical speed to keep it centered whilst continuing to intercept the LOC.
In all aircraft I've flown (light aircraft & turboprops), the autopilot won't "capture" the GS first - but the pilot is quite likely to manage the flight path to achieve the same goal.