Can I use USB A to USB C for Migration Assistant?

Transfer to a new Mac with Migration Assistant - Apple Support

Transfer files between two Mac computers using target disk mode - Apple Support

For Target Disk Mode: "If either of the computers has macOS 11 or later installed [which the new MacBook Air will], you must connect the two computers using a Thunderbolt cable." Here we'd be talking about a TB 3-to-2 adapter plus a Thunderbolt 1/2 cable. So migrating from an up-to-date external backup drive might be the best thing to do.

I'd suggest migrating from an up-to-date Time Machine or bootable clone backup on an external drive – rather than over a USB cable between the two Macs.

If you need to use a USB-C to USB-A adapter to connect the drive, no problem.

Yes, will be slow though...

Transfer files between a Mac with Apple silicon and another Mac - Apple Support

Or if you have a Time Machine Backup you could move that over...

Transfer your data to your new Mac Pro – Apple Support (MY)

The 2015 uses thunderbolt 2. There are thunderbolt 2 to 3 adapters. You can also link the two via ethernet and a thunderbolt 2 to ethernet adapter. USB-3.0 only supports migration assistant from external hard drives, and no target disk mode.

As

above I have a Mac Air 2013 Catalina platform to new 24"Mac 2024.

Can I use a USBC to USBA cable.