Flatten multiple documents at once

Do you mean you are adding stamps? and you want to flatten the stamp comment into the page content? If so, there is a preflight fixup for that. Tools> Print Production> Fixups (blue wrench icon). If you want to apply this preflight action on multiple files at the same time, you can do this by creating a new Action with the Action Wizard tool, Tools> Action Wizard> New Action> Document processing> Preflight> Flatten all annotations into page contents*> (uncheck "Prompt user" to run the preflight on all files in a folder without interuption). > Add a Save tool (+) to your new action and save it.

With a PDF open, go to Tools> Action Wizard> select your new Action from the Action List, where it says "Add Files" click and change it to "Add Folders" and navigate to a folder full of the PDFs you want to flatten. Note, you can change the saved file name and folder destination by editing the Action.

*Tip: If you don't see a desired preflight profile in the Action Wizard, you might first need to export the profile to your desktop from the small fly-out menu, then import it back into the preflight profile panel, renaming as needed.

If the suggested preflight profile is not what you want, you can apply pretty much any preflight profile to multiple PDFs in a folder using the same technique.

Assuming you want to flatten form fields and annotations:

Exactly what do you mean by flattening. That term can mean multiple very different things within the context of Acrobat and PDF files.

The most common use of the term flattening involves taking form fields and annotations and merging their contents into the underlying PDF content. After such flattening, the form fields and annotations are no different than regular PDF content.

The second most common use of the term flattening is for “transparency flattening” in which items that are not totally opaque are blended with underlying content to yield PDF objects that are totally opaque.

Finally, some folks misuse the term flattening to mean that all content in the PDF file is turned into raster images in a very mistaken belief that this will prevent any modification of the content (it doesn't!).

There is no built-in function to perform these functions on multiple files. Conceivably you can use the Actions function to put together an action to do such operations on multiple files, but the operation of such an action will not be any less time-consuming than doing them manually. Any actions are serial, one file at a time!