It gets more complicated in larger stars of course. Stars similar to our Earth run in similar patterns. Massive stars that create super giants at the end cycle don't follow this pattern exactly, they actually end up with fusion "bands". Kinda like an onion has layers, with different elemental fusions happening at various levels of its shell.

Red Dwarves, the smallest celestial objects that can be considered stars may not have this expansive property much at all. We don't actually know for sure. The smaller a star is, the longer it lives for. Red Dwarves are so small, and live for so long, that the lifetime of a Red Dwarf is presumed to be longer than the current age of the universe. We can theorize a bit as to what happens but we have absolutely no means to observe an end stage red dwarf. The universe, presumably, isn't old enough for that to have occurred yet.