Clean the chamber really well. Then go back and do it again. Check your extractor. Is the hook chipped? Does it move under its own steam to its most closed position without assistance, and while youre there does its movement feel sticky? Mini-14's reputation for reliablilty is good with factory mags. Without actually seeing it, it sounds like the extractor has insufficent grip on the cartidge rim, and when the gun cycles(and the Mini cycles hard) the extractor is popping off the rim. When you manually cycle the action (more gentle cycle) the case come out.
so: Dirty chamber (lacquer left behind by cheap ammo, etc?)
Chipper extractor
limited/weak range of motion on extractor