This appears to be a frequent question on the web, including myself. Well I'm happy to share with you the answer I found. I tested it May 2014 on the latest IOS and it works great.
I believe this may be be the developers shortcut because it's not an obvious procedure nor does it produce prompts or confirmations. I will add my own tips here and reiterate those of the original poster:
1) make sure you are still holding the "move" button while you unselect the initial email.
2) be PATIENT. Allow the list to build before selecting the Trash or other folder. Hint. The folder links will darken once the OS has finished selecting the files. Another hint. Once your have selected the desired folder, I.e. trash, WAIT. This process will also take a time depending on the number of files but it will work.
The first time I tried this, I crashed the mail app by being impatient and tapping links, etc. during the process.
Good Luck!
"How to delete ALL mail messages from iPhone/iPad in one step
by Adrian Segar
Yes, there is a way to delete all your unwanted iPhone/iPad emails from the Mail app in one operation! No more left-swipe:tap Trash for every individual message. No more Edit: tap the single open circle next to every individual message and finally tapping Trash. And you don’t need to jail break your device.
If you leave your iDevice on for a few days and come back to find a few hundred messages on it that you’ve already downloaded elsewhere this trick will save you time and irritation. I didn’t discover the method—it’s far from obvious—but found it on one of many Apple discussion threads bemoaning this irritating hole in Mail functionality.
It works! I present to you this great tip from shashbasharat found on MacRumors (slightly edited for clarity).
How to delete or move ALL emails at once in non-jail broken ipad or iphone
It took me weeks of research to figure out finally how to decode this yet another secretive secret of apple. There is a perfect way of deleting ALL emails at once without jailbreaking your iphone or ipad…and here it is:
>> Open Inbox >> Edit >> Check/select the top message; it will highlight the move button.
>> Press and hold the move button and uncheck the message that you had checked earlier.
>> Lift all your fingers off from the ipad screen and leave it alone. Wait until all your messages pile up on the right hand portion of the screen (in ipad); iphone will give you the actual number of emails it has selected for the action. Now they are just waiting for your command to be moved ALL at once.
>> Choose trash to delete all of them or any other folder where u want to move them, like one piece. Remember this will replicate your action on the server so you will ACTUALLY move them or delete them on the server and not just the ipad.
Once you have moved all messages to the trash you can either leave them there for the scheduled cleaning or empty it right away by doing this: go to trash folder and touch Edit. The Delete All button shows up at the bottom of the screen. Hit it! You’re done!
If you do not see the effects of your actions on the server make sure you have enabled your email accounts for such actions.
Tips:
1. Please give enough time (could take up to several minutes depending the number of emails to be moved) for the emails to be selected for the move. Your screen will become unresponsive while all emails are being packaged. Once emails are ready, in ipad, you will see them zoomed out on the right hand side of the screen, and in iphone you will see the message showing you the actual number of messages that have been selected.
2. Avoid purging very large number of emails, the mail app might freeze or crash. If your inbox has thousands of emails change your sync settings in mail settings to fill lesser number of emails in your inbox"
