Vessels for selectively holding drilling cuttings material and methods employing such vessels; the vessels, in at least certain aspects, having a body with a hollow container with an interior space for receiving drilling cuttings material and from which the material may exit from the body; a first opening through which the material enters into the container; a second opening through which the material passes out from the container; and movement apparatus with a movement member within the container movable adjacent the second opening to facilitate passage of the material into the second opening; and in certain aspects, the vessels having moisture-content sensor apparatus on the body for sensing moisture content of material within a vessel; conduit apparatus for conducting drilling cuttings from the exit opening; valve apparatus for selectively controlling flow of material in the conduit apparatus; and a controller in communication with the moisture-content sensor apparatus and the valve apparatus for selectively controlling flow in the conduit apparatus.