long as you have enough liquid in the pot, you can stew and braise on the stovetop nearly as well as in the oven. The benefit of the oven, of course, is that the heat reaches the pot from all directions, while the heat of the stovetop is concentrated on the bottom.
The thing you have to look out for, then, is that the food on the bottom of the Dutch oven doesn’t stick and burn. Make sure the heat is low enough, the contents of the pot liquid enough, and that you stir often enough so that this doesn’t happen. If you keep an eye on things, it shouldn’t take much longer to cook your dish than in the oven.
There are pots on the market with synthetic (plastic) handles that are designed to withstand the heat of the oven. You might want to check with the manufacturer to see if yours is an oven-proof pot in disguise (but don’t try it unless the manufacturer approves).
Related Article: Explain the Dutch Oven to Me