Some things I'd be looking at trying is:
---> Reset ACL's
Boot to Recovery -> Utilities -> Terminal
Then enter the following command to open the Reset Password Utility:
Then select the volume (normally Macintosh HD)
Choose the user account in the "Select the user account:" pop-up menu
Click Reset (ignore all the 'enter new password for user' sections and just click reset at the bottom after selecting the volume and the user
Once that is done, quit Terminal and the Reset Password Utility
Open Disk Utility
---> Verify/Repair Disk + Permissions
Click Macinosh HD on the left of Disk Utility and then click Verify Disk at the bottom right, if it comes up with green text in the white box saying 'The volume Macintosh HD appears to be OK', then you DO NOT need to click Repair Disk, if it finds any errors, you should click Repair Disk
Afterwards, you should click Verify Disk Permissions, once that completes click Repair Disk Permissions (Click this regardless of what Verify Disk Permissions says)
Once that is done, I would follow HT2409 and do a Spotlight Re-Index
Attempt another backup and see how you get on
@Eagle Owl: You are mistaken. They are not meant to be locked.
None of mine are locked, if yours are, then there might be an issue.
@GaryKing Could you take a screenshot of the Info window for your Time Machine disk? (Command-I)
Also do you have anything else on your backup disk? Are any of those locked?
Maybe this site can help? http://pondini.org/TM/Troubleshooting.html
If it helps, when I restored an older Firefox profile, and ran Firefox, I got an error saying that the files were locked. So I checked the folder, and the locked files copied over, so that's really problematic because I had to open Terminal and recursively unlock all files in the folder. That's not a good thing.
I have nothing on the backup disk besides Time Machine. And in fact, this drive is fairly new; I copied over 2 TB from my old backup drive to this new one, a few months ago. And so I guess to answer your second question, I have nothing locked besides my Time Machine backup files, and I don't know why those are locked.
I just tried "Reset Home Folder Permissions and ACLs" and it looks like it might be helping. I ran this for about 30 minutes, canceled it because it took too long, backed up Time Machine again, and noticed that Time Machine backed up some files that I did not touch. Turns out that these files were the same ones being locked in backups, and in the latest backup, they were no longer locked.
So perhaps I was having permission problems with the files that were getting locked? I don't know if this "fix" will resolve this permanently, or just for now, but it certaily seems to be related.