I've gone into the Kremlin with normal sized bags without any problems. I think you walk through a metal detector with belongings being X-rayed (or do some sort of alternative security check) at the entrance at the wall [I remember a sort of little bridge and gatehouse built into the wall where we entered and the security checkpoint was in that gatehouse), but once inside the area, I don't recall any further security or restrictions for individual buildings such as churches and museums.

However, when I visited Lenin's "tomb", you absolutely could not take in any cell phones (and could not chew gun or do anything that might be considered disrespectful). It's probably changed, but we were sent to some shopping mall place a block or two away (which was hard to find) where there was a place to store cell phones and other stuff that couldn't be taken into Lenin's tomb. It was a strange arrangement. Lenin's tomb also keeps very limited and idiosyncratic hours; there can be long lines even through people are required to move through fairly quickly and not loiter in the chamber with the body.