This new fuel will only burn at a higher ignition temperature requiring further contraction of the core. Eventually shell burning can't release enough energy to support the star against gravity.
Gravity compresses and heats the core.
This step is the same as the first one above. This starts a new cycle with a new fuel. The star will continue through this cycle until degeneracy pressure takes over and prevents the star from contracting further and thus heating the core so that it can burn the next fuel on the list.