To me, when I hear "dumping," I'm thinking the person wants to trade or DFA the player for whatever the GM can get, no matter how bad the return is, or no return, if DFA'd.
Nobody wants to "dump" Verdugo, Duran, Casas, Mayer and many other players or prospects mentioned, recently. Trading any player with value should return close to equal value, so it would not be a dump. Often, these trades involve getting a player for a position of higher need or trading a player with limited team control (and may not be in the long term plans) for a player with longer control. The concept does not seem hard to grasp, yet so many jump all over anyone who suggests a player or prospect they think might need to be part of a package to get a player we may need more.
Other factors involve who we have for depth at the position we want to trade. If the drop off is not much, we can basically stay close to even at the position we trade from and gain more at the position of higher need. For example, if you think Duvall (re-signed) + Abreu is not a big drop from Dugo, but we can get a decent pitcher for Dugo, how is it "dumping him?"