I was wondering how studios handle render passes when a Houdini FX artist is asked to create effects for an animated object like a car or a character, e.g. adding particle effects. Are the character objects generally rendered inside Houdini as well, or in another 3D app like Maya to combine later?

Is a character, for example, usually animated in Maya and then passed to Houdini to render in Houdini?

Someone working in a larger studio can probably tell more you than I can, but we do a little bit of everything depending on the job.
Most commonly we export the models and parts we need as Alembic or FBX into Houdini, create the effects and use the scene/models as matte objects. This works fine for most pyro and particle effects. For some shots, it's just easier to do the full rendering in Houdini. And in some instances we export our Houdini caches back into Max.

Tronotrond
Someone working in a larger studio can probably tell more you than I can, but we do a little bit of everything depending on the job.
Most commonly we export the models and parts we need as Alembic or FBX into Houdini, create the effects and use the scene/models as matte objects. This works fine for most pyro and particle effects. For some shots, it's just easier to do the full rendering in Houdini. And in some instances we export our Houdini caches back into Max.

Thanks for your helpful reply.

So by example, if I have a character coated in a water effect and yu just want to render the water, do you make the character a matte so that the water renders as transparent so it can be latr composited on top of the character?

Also, regarding FBX, can this format be used to bring in the textures (unlike alembic)? Is this how you would render them together?

Lastly, by "caches", what type of cache do you mean? I think I have seen a reference to a "geometry cache" -- is there such a thing? If so, is this like exporting a mesh, say of a water sim?

It's different in every studio but that's why standard file formats like Alembic, VDB, USD exists. Even though you can technically render Deep images out of Houdini and comp them with other Deep renders from another software, things never line up properly and you'll get issues with mattes/edges - and you will need interaction between 3D elements anyways for shadows, transmission, reflection, etc...
You should render everything in one software, whatever it is, Houdini, Maya, Katana, Gaffer, or others.