My newish MacBook Air was just lost to a coffee spill, and I need to migrate back to my MacBook Pro using a recent back-up (Time Machine to external disk). Can I do this?

You certainly can safely migrate your Home folder. Just don't try to do a full disk restore. You can also manually restore specific folders if you prefer. This may be faster since you will be restoring less data, and not using Migration Assistant.

I suggest you don't try using any Migration type software. It really isn't meant for going back to a older version of OS X.

What you want to do is just copy over your Office docs, whatever they are, and other personal files you may have.

Do not attempt to copy files/folders from a Time Machine backup. Only use the Time Machine application to restore from the backup.

Click on a Finder window and select Help from the Finder's Help menu. Search for "restore" and select "restore items backed up with time machine" from the hints list. You will find these two articles:

Recover items using Time Machine and Spotlight

The restored item is returned to its original location. For example, if the item was found in the Documents folder, it’s returned to the Documents folder.

Restore items backed up with Time Machine

If you use Time Machine to back up your Mac, you can easily bring back lost items or recover older versions of files.

You can use Time Machine within many apps. If the iCloud Documents & Data feature is turned on, you can recover older versions of iCloud documents as well as regular documents.

1. Open a window for the item you want to restore.For example, if you accidentally deleted a file from your Documents folder, open the Documents folder. To recover an email message, open your inbox in Mail. To recover an iCloud document, open the document’s app, choose File > Open, then click iCloud to view the iCloud documents for that app.If you’re missing an item from the desktop, you don’t need to open a window.
2. From the Time Machine menu in the menu bar, choose Enter Time Machine. A message may appear while your Mac connects to the backup disk.
3. Use the arrows and timeline to browse the snapshots and backups.White tick marks represent snapshots stored on your internal drive (portable computers only). Pink tick marks represent backups stored on your backup disk. If a tick mark appears dimmed or grayed, there’s no backup available or the backup disk is not connected.
4. For more information about an item, double-click it.The windows in Time Machine behave just like Finder windows, so you can open folders, click items in the sidebar, and use the search field in the upper-right corner of the window.
5. When you find the item you want to restore, select it, then click Restore. You can restore individual items, multiple items, folders, or your entire hard disk.The restored item is returned to its original location. For example, if the item was found in the Documents folder, it is returned to the Documents folder.

If you become confused then:

Please visit Pondini's Time Machine FAQ for help with all things Time Machine.

Essentially, for now all you want to do is restore the files you need to your Pro from the Time Machine backup.

Mac Basics- Time Machine

Time Machine- How to transfer backups from the current backup drive to a new drive

How can I see my backups, or the backups for a different Mac?

How can I copy my TM backups to a different location?

How can I restore a file/folder to an alternate location?

When I said to copy your files over I was not talking about from any Time Machine backup. Although I have directly copied files off a TM backup without any problems.

I was suggesting to connect the drive from the liquid damaged system to the other with a SATA to USB adapter of some type.

Sorry for the confusion and not stating that in my other post.