Git does not record moves explicitly (like for instance SVN does). Each commit is a snapshot of the entire state. So, it does not matter if you moved the file using git mv A B or moved it using mv A B && git add A B, or moved it in any other way and added this change -- the result will be identical.
Git instead provides tools to view the history in various ways. Whether or not a particular change was a rename is determined on-the-fly as the data is viewed.
Using the --follow option to git log will enable rename detection.
git log --follow -- path/to/file
Sometimes when a file is renamed, one might also change its contents. Depending on how much of the content has changed, rename detection might not work -- it becomes the question of whether it is a rename, or a deletion and an unrelated addition. To change how much of the file that needs to match for it to be considered a rename, use the -M option.
git log -M40% --follow -- path/to/file
The -M option also works with other commands such as git diff and git show.
There is also -C which is equivalent to -M except it detects copies.
If a file needs to be renamed and its contents also change a lot, a trick is to, if possible, do the rename and the content modification in separate commits. That way the rename will be detected without the need for tweaking.
More info at man git-log, man git-diff, man git-show.
