Thermally protecting a varistor
One of the main differences between a TVS and a Varistor is that the voltage tolerance of a TVS is much tighter than a Varistor.  They also don't appear to have the "wear" factor that Varistors have.

Varistors that are repeatedly subjected to transients appear to have their clamping voltage lowered as a result of the stress.  Eventually, they become conductive at the operating voltage and then fail catastrophically.  That is: they get very hot, then turn black, then eventually catch fire.

Many of the better AC power bars that have Varistor clamps within also have a thermal cutoff device mounted between two Varistors.  The thermal cutoff device disables the power bar when a Varistor fails.

In terms of what happens to a transient suppression device when it is exposed to voltages in excess of their rated clamp voltage: they get hot.  The will, in fact, get hot enough to fail.

When a Varistor fails because of over-temperature, it generally becomes restively-conductive.  That is, it turns black, then catches fire.  Your over-current device may or may not operate, depending on how low the resistance of the Varistor has dropped.

When a Tranzorb fails because of over-temperature, it generally becomes a short-circuit.  This causes sufficient current to flow to cause your over-current device to open.