Facebook uses what you call, db caching servers, which uses memory to hold data. It is way more complex that what I am describing but instead of doing a lot of mysql db reads they mostly only write to them unless a caching server reboots it will rebuild its data back from the mysql backend servers. This causes a lot less load on them as well as spreading and duplicating it over multiple servers in memory makes the whole process faster for users and servers. I believe the specific product they use for this is memcached if you want to read up on that. Obviously they do the same for the webpages as well.
However most things facebook are not replicated all over the world, they typically will have 3 data centres in different locations that have the web servers and web caching servers as well as with the mysql and memcached servers. The only thing that is local is Akamai, which is a CDN and most of the time this only serves static content, ala pictures, javascript, stylesheets ect, the dynamic content will still go international regardless.