What kind of control does the FX's ATtiny have over the ATmega32u4?

This is a specific question targetted specifically at those who are privy to how the FX chip is designed.

There was a recent question about setting the Arduboy’s lock bits and it got me thinking:
does the ATtiny installed on the FX chip upload to the Arduboy’s ATmega32u4 by sending bootloader commands over SPI or does it actually manipulate the hardware in the way that a USBASP/ISP would?

I’m mainly interested in that one specific thing, but I’ve made the question slightly more general because I think any other information about the relationship between the two is likely to come in handy.

You mean the FX mod chip. The FX chip refers to the flash chip only. Also the attiny will be available only on the FX mod chip and Arduboys upgraded with the FX mod chip. Future Arduboy FX with new PCBs will not have an attiny (unless @bateske changes his mind)

Yes you can think of the attiny being a USBASP/ISP (but without USB part). it functions as an ISP programmer and will program Arduboy with the fuse settings and hexfile embedded in the attiny.

When commanded to program Arduboy (by Arduboys request or long pressing reset)
it will do a chip erase the atmega32u4 (which also resets the lockbits), sets the fuse settings and then programs the Cathy3K bootloader.

Fuse settings and firmware can be customized by adapting the FX mod chip project. But only is only recommended for advanced users who know what they are doing.