I used a Nexus 7 for a couple of years and got a Nexus 10 earlier this year. I installed a number of kids books/games on the 7 and wanted to transfer them to the 10. Now I can't find some of them at the store. Would the APK files still be located on the 7?
Try ES file explorer
Under settings select App Manager and select all apps. And select backup on the action bar below.
A backup of all apps apk is created to your SD card probably under backups folder(or check /set backup folder on your es file explorer, settings-> directory settings )
Now you've got all apps apk.. You can either transfer them through Bluetooth/Wi-Fi /USB connection
You can select system apps too if your phone's rooted
I strongly recommend using AppMonster (or rather AppMonster Pro) not only for this issue, but in general. It can create backups of all your installed apps, and the Pro version even offers to do so automatically on each install/update. That way you cannot only transfer
.apk files to another device (as in your current case), but also have a fall-back available in case an update breaks stuff (or removed features): As AppMonster keeps multiple backups (one per install/update), you can easily revert to any version previously installed. Saved my day more than once.
Everything you've bought or installed from the Play Store is listed under your account. A lot of the time apps that have been withdrawn from the Store are still visible to people who've used them before in this list:
If you look down that list then you should be able to see the apps that you've installed before and push them to any of your devices.