I have an old phone with a broken power button. I'd like to transfer the app data (emulator save file) onto another windows phone. Is that possible?
You can transfer your app data to an another phone, but using cloud. Here is how you can do it.
These backups will be restored while you install apps on the new device. Make sure both are connected to the same Microsoft account.
Note : This method does not work for all apps as it is upto the devs to implement it.
For unsupported apps
There is an unofficial way to restore app data for apps that don't support OneDrive backup. In this way apps are moved to SD card and their app data is edited. I am not familiar with such things. You can refer to @CBHacking's answer here in which he says,
There are options, but they're all terribly unofficial and usually somewhat "hacky". Both getting your files off the old phone, and getting them onto the new one, are official not possible. For the old phone, there's actually a lot of options (so long as you're still on WP8.1): use an interop-unlock and capability-unlock hack to browse the file system to the isolated storage folders for apps using an app on the phone, use a "full FS" hack for MTP and do it via USB, use http://www.wpinternals.net/, or put the apps on SD card and use on-phone hacks to get at the app data on the SD card. I can explain these in more detail, but you'd be better off checking sites like http://forum.xda-developers.com/windows-phone-8/development for tools and instructions. Getting the extracted files back onto a 650 is going to be harder. While W10M is actually more permissive than WP8.1 when you're working without hacks, it's harder to hack (or at least, we haven't gotten as far yet). SD card hacks may be an option (see the Lumia 950 forum on XDA; most stuff will work on the 650 as well).