How to create product operator for arbitrary binary operator

Question

I have defined a couple of different mathmode binary operators in my paper, but I realized I need to apply them to arbitrary finite sets of objects. Is there a standard way to make a related product-like operator, which is bigger and has displaystyle limits by default? I want the new operator to have the same relationship to the old one that \bigoplus has to \oplus.

Ideally, I am looking for a way to define a command like \bigbin{} which takes a hypothetical binary operator \newtimes in the argument and produces a product version. So \bigbin{\newtimes} would look the same as \newtimes, but it would be bigger, and have display-style limits. Failing that, I would like to know both how to make an arbitrary glyph bigger, and what it is that makes the product operator special, besides display style limits—i.e. how to make my own \bignewtimes command looks like a bigger \newtimes but acts like a product operator.