If you're moving or transferring files from one user to another within a Google Apps domain, that can be accomplished using a transfer tool within the Admin Console.
And the bad news: if you're moving or transferring files from one user to another outside a Google Apps domain, there is not a native tool for that function.
However, there is a work-around. Here's how it works.
Let's pretend you have user@domain1 and user@domain2.
You want to move (change ownership) of files from user@domain1 to user@domain2.
In the user@domain1's Drive, create a folder called INBOUND and share it to user@domain2.
In the user@domain1's Drive, copy or move all of the content you want to move into the INBOUND share.
In the user@domain2's Drive, find the INBOUND share and make a copy the file.
The copied file is now owned by user@domain2.
Unshare the INBOUND folder in user@domain1's Drive.
Now, all of this sounds pretty cool until you notice two things:
1. Google Drive doesn't have a native way to copy a Folder. What an oversight?
2. Google Drive appends a text string of "Copy of" in front of the file. Very obnoxious.
Low, another work-around.
In user@domain2's Drive, create a new Spreadsheet.
Add an Add On called copy folder. You'll need to search for it and grant it permissions.
In the spreadsheet, select the Add On copy folder.
The UI is pretty simple: select the INBOUND folder and process.
The script will recurse through the INBOUND folder and make a duplicate of it; it will also recurse the subdirectory structure and re-create subdirectories.
The copied INBOUND folder will now be owned by user@domain2.
You can now unshare the INBOUND folder.
A couple of notes on the copy folder add-on.
It's not perfect. Expect it to crash on long recursive processes.
I think the "breaking" of recursive processes may be due to ownership rights of objects within the folder being copied - but I'm uncertain at this point.
Instead, try to break down your copying process into batches - handling sub-directories separately.
One day, maybe ... soon? ... Google may feel nice enough to create a means of copying a Folder within the web user interface. Then, one day, by the Grace of Google, we could assign ownership rights outside of a Domain to another Google Apps user. But until then, I offer these work-arounds.