Which component's video processor takes precedent?

What processing? There are so many things that fall under that umbrella. It all depends on what you're trying to accomplish, what components you have and what their instruction manuals say. If several components in your signal path do the same thing, deinterlace, for example, you need to figure out which does a better job and configure the components so that it does the deinterlacing. Others, like the Darblet, do unique things and there's only one place to do that, with the Darblet. So, start at the other end. Figure out what processing you want and then look at your components to figure out how best to accomplish your goal.

The simple answer is that whichever component does the best job is the one that should do all the heavy lifting, and you should turn off processing everywhere else. For example, if you've purchased an outboard standalone video processor, chances are that you expect it to process the video. By "process the video," I mean deinterlacing and scaling, and possibly calibration and/or aspect ratio control. If you bought an external VP for that, you wouldn't want to duplicate that effort in your Blu-ray player, receiver or other piece of hardware.

If you haven't bought an external VP, you probably need to do a little research into whether your Blu-ray player, receiver or TV has the best processing, then arrange things to use that.

An exception to this rule is the Darblet, which is a specialized component with a unique feature. The Darblet works in conjunction with other video processors, not as a replacement for them. It doesn't deinterlace, scale, calibrate, etc. The one and only thing it does is apply the Darblet effect to give your image a little extra "pop" after calibration is complete.