How to XOR automata?

Say we have 3 DFAs. We know how to OR, AND, or NOT them. But how does one XOR them? There is not one single mention of this online.

$x\; \mathrm{XOR} \;y\; \mathrm{XOR} \;z = ((x|y)(\neg x|y)|z) (\neg ((x|y)(\neg x|y))|z)$. This is way too complicated and time consuming to draw. Isn't there another way?

Thank you for taking the time!

Note that the construction for the intersection and union ("and" and "or") of two automata is exactly the same, except for the definition of which states are accepting. The same principle applies to any Boolean combination of any finite set of languages: use the product construction and the appropriate definition of which states should be accepting.

StackExchange Computer Science Q#33018, answer score: 10