Player improvised battle terrain

Often, Especially in the theater of the mind, in combat a player will ask something like, "is there a barrel in the corner for me to hide behind?" And the game master Will say something like, "yes and you receive three quarters cover." In an even better scenario the player might asks, "can I push over the barrel so that the monster slips on the pickle juice?" And the game master will say something like, "Give me a strength check to topple it over, and if anybody moves through the pickle zone they must do a dex check or fall prone." While this is fun and dynamic play often, I imagine, players will be either too shy to ask or too set in their ways to improvise and think outside the box. If there was a simple mechanic or convention to encourage this type of on-the-fly Battlemap generation I think it would be good. For example, maybe directly after rolling initiative one player in the rotation would be encouraged to put something on the map, with whatever limitations or lack thereof the game master desires (maybe the person who's last in initiative order as a consolation prize). Or maybe this could be tied in to the inspiration mechanic as a way to spend your inspiration point from last encounter. At first I considered going around the circle and having each person put one thing on the map but, after a little thought, I wonder if that would bog down the progress of the encounter too much. The main drawback with using a solidified rule that I foresee is having players be intimidated to be ad hoc about things. Not sure how this will play out, just some food for thought.