I just play off cross class abilities as the class being able to do something that's just like what another class does, but isn't necessarily tied to that class. Some cases (like death blossom) are tricky to explain, but I figure cross class abilities are basically just one class using its own skills to do what another class does. Let's take BLM (the only class I really play) as an example:
Raging Strikes - wizard gets angry, makes bigger boom
Quelling Strikes - wizard casts spells in a sneakier fashion/starts crying so enemies feel sorry for them
Eye for an Eye - we can ward ourselves, we can ward others! it's like apocatastasis!
Virus - spooky evil curse, lore says we can do those
Physick - wizard band-aids
Raging Strikes - wizard gets angry, makes bigger boom
Quelling Strikes - wizard casts spells in a sneakier fashion/starts crying so enemies feel sorry for them
Eye for an Eye - we can ward ourselves, we can ward others! it's like apocatastasis!
Virus - spooky evil curse, lore says we can do those
Physick - wizard band-aids