Determining font for a given Unicode codepoint

Is there a specification that details how a browser should determine which
font to use for any given Unicode codepoint, or is it up to vendors to
determine it themselves?
My understanding is that a browser will run through the fonts specified in
the font-family declaration until it finds one that has that glyph. If not
it looks at its own preferences, and then a system fallback font.
I'd like to understand why the same page on the same machine would support
Unicode characters differently in two different browsers, if both could use
the same system fallback font.

Thanks
John