IMHO I think Mowbray has far more tactical nous than the likes of Pulis. It's just immensely difficult to play a passing game outside of the top 6-8 clubs and stay up.
Then if you have 'tactical nous', you realise you are not top 6/8, and change the gameplan, and that
may mean getting rid of the players who took you up from the the league below. Stoke were favs to go down, but survived comfortably, to say Pulis had less nousthan Mowbray flies in the face of all evidence.
And if you've not got money to replace them?
then you try to play a different game with the players you have.
That's what he'll need to do at celtic as he won't have the cash to buy decent players that can pass the ball properly.