I have a question: what is the difference between a local version of Google vs. the default Google in regards to search results?
Many factors, the results in google.com can be a real mix of international listings as well as local. The location of backlinks to a website can force Google to rank a site in that region.
Lets say you have a .mx site but have lots of links from UK websites because it is about holidays in mexico then you could rank well in the google.co.uk it would be more likely however if the domain was a .com as Google gives more universal power to those types of TLD's
The domain is a .mx site, so wouldn't it make more sense that this page would rank higher on google.com.mx instead of the default Google site
Not if the above is true, also if you have a less of a local feel to your content you might rank better among a group of similar sites that are more broad in nature.
Also, what determines whether a user gets automatically directed to a local Google version vs. staying on the default one?
There are a lot of factors involved, IP address and Google's ability to determine where you are.
The language you use and even the spelling of words, for instance in the UK vs US it could be color vs colour
There are lots of answers to your questions but those are just a few to give you an idea of the 200+ algorithm factors for ranking in such circumstances.