Transfer photos to save them off of an old Mac 2010 desktop.

Have you put a lot of organizational effort into your Photos 2 library, i.e. folders, titles, keywords and descriptions?

If you haven't the do this:

1 - copy the library to an external drive formatted either APFS (if SSD) or OS X Extended (journaled) if HDD.

2 - plug the drive into your new Mac, open your Photos library and use the File ➙ Inert menu item to select and import the photos from your old library into the new one.

If you've put a lot of organizational effort into the old library this:

1 - copy the library to an external drive formatted either APFS (if SSD) or OS X Extended (journaled) if HDD.

2 - plug the drive into your new Mac.

3 - purchase a copy of PowerPhotos .

4 - open PowerPhotos and add your old library and your current library to it's sidebar.

5 - merge the old library into the new library. This is a user tip on the differences between using Photos vs PowerPhotos: Importing a Photos Library into Another Photos Library: PowerPhotos vs Photos - Apple Community.

The year these various devices were made/purchased is of less relevance that the version of the operating system on them, and well as the actual software used to manage the photographs.

Most basic solution: Start up the Mac and connect a flash drive or external drive to it and drag the images to that drive.

This could be more complex if these images are in a iPhoto or Photos library. Are they? And of so, what version of the OS is on the Mac, and where are you hoping to store them going forward?

To help we need information from you, and frankly, I can't see the connection between the title of your post, the body of your post and the equipment you mention. Can you clarify?