In UBW, when Rin removes Shinji from the Grail, the Grail attempts to take him back (with little hands). Then it grows a massive hand, but instead of going for Shinji, it goes for Gilgamesh to replace its core. Why didn't the hand try to retake Shinji, if he was much closer than Gilgamesh?
One could argue that Gilgamesh would be much more fit to be the Grail's core, but I'm not sure if that's explicitly mentioned or not. Besides, Shinji's body was doing the job just fine I think.