An ideal mixer is a classic example of a time varying linear system - from the input port to the output port, not the LO port. If you mix a signal A with the LO and get out signal X and you mix a signal B and get out Y, then mixing A+B will give you X+Y (in an ideal mixer). This establishes linearity. But, if you mix in Acos(wt) and get Xcos(w't + phi) and mix in A cos(w(t+dT)+phi), you will not in general get Xcos(w'(t+dT) + phi). This makes the mixer time variant. A simple multiplier is an LTI system, a mixer is a Linear Time Varying system.