Question: Is there a good way in Perforce to move unsubmitted changes to a different stream before submitting them (equivalent of `git stash`, `git checkout otherbranch`, `git stash pop`)?

I currently have the files checked out locally (non-exclusively) and edited. I've tried shelving them and then trying to unshelve them into the target stream, but I get "file(s) not in client view". When I view the changelist (shelved or not), the files all have paths that include the original stream.

My target stream is one I just created, parented off the original stream. I'm using P4V. The version of Perforce Visual Components I have installed is 123.57.9578, and when I run p4 from the command line, it says "Server 2012.2/551823". I can add more info if necessary.

Answer: 
The other answers here are correct, but they don't warn you that you can't just shelve from any stream and unshelve on any unrelated stream. In particular, the original poster asked about the error message "file(s) not in client view" which is the error message that P4V displays when you haven't selected the proper mapping.

Here's the trick to finding the proper mapping: Your unshelve stream must have a direct parent/child relationship with the shelved stream, and you must select the stream spec that describes that relationship in the "Map unshelved files using stream..." part of the dialog. This means it may take multiple hops to get from the original shelved stream to the target stream.

Here's a concrete example: Let's say you have files shelved into `Dev-1` that you want to move to `Dev-2`. These two Dev streams are both direct children of `Main`. You can't unshelve `Dev-1`'s files directly into `Dev-2`, first you have to unshelve and temporarily re-shelve into `Main`, because the Dev stream specs both describe their relationship in terms of `Main` and not each other.

So, you unshelve `Dev-1`'s files into `Main` with "Map unshelved files using stream `Dev-1`" because that spec describes the relationship being traversed. Then, shelve the files into `Main`, and then switch your workspace to `Dev-2`. Now, you can unshelve the `Main` version of the shelved files, with "Map unshelved files using stream `Dev-2`", because that spec describes the direct relationship from `Main` to `Dev-2`.

With multiple hops, you can get to any other connected stream in your network. This is not as easy as `git stash` and `git stash pop`, but remember that Git is assuming the filenames don't change between git branches, whereas Perforce allows each hop to arbitrarily change the mapping of files and folders.
I believe the shelve/unshelve technique requires the 2013.1 server, not the 2012.2 server. You should confirm this with Perforce Technical Support.