If travelling with luggage in SE Asia and not staying in hotels, where can I leave my belongings?

I'm wondering whether it is possible to live in hostels, Airbnb and bungalows and yet still be unencumbered with a "rollaboard" type luggage one might bring along

Typically, do these places allow you to store luggage with them as you travel during the day?

If not, are there any other places that could do this for you?

Hotels could probably hold your luggage but they are also more expensive to stay in, so this question is geared toward the budget-minded, and the (not so light-packing) backpacker, who might be moving from city to city very often.

I realize a backpack is probably preferable, but if you have any sporting equipment, camera gear, or other heavy/expensive belongings, then there are times you just want to leave these things wherever you are staying. So bringing a backpack might be lighter, but you might end up leaving it at the hostel anyway. (If you have luggage, there's no reason why you can't throw some items into a small daypack when you go out.)

Do any of the budget minded places I mention offer to hold your luggage?

If so, do they all offer it so that you don't have to ask each one every time you move, or can you simply assume they can?