An excellent verb is "rummage". It is typically used with the prepositions "through" and "around" accordingly, or simply with "in".
It should be noted that only the context suggests that "rummage" is without the owner's permission. It does not directly mean that.
Another viable suggestion is "rifle". It is used with the same prepositions as above. This verb is more adept to stealing/looking without the owner's permission. Here are some examples to suit your ideas:
When I turned around, I saw somebody was rummaging through my bag.
My toddler nephews like to rummage around wardrobes and bookcases.
I came home to find a masked man rifling through my stuff.
I couldn't believe my eyes when I saw the security guard rummaging through the piles of suitcases.
Another option is "ransack" though this carries an air of "pillage" which has an air of violence about it. This is perhaps not the preferred option if you wish to express purely "without the owner's consent/permission".
We had a call from the police to say the summer house had been ransacked.