I want to export all the information from my old Mac (the circular one) to a ThunderBolt 4 hard drive.

Each and every new drive needs to be Disk Utility ERASED by its immutable manufacturer-given Device-name before it can be used on a a Mac. Be sure to select "show all devices" from the tiny show menu.

Erase and reformat a storage device in Disk Utility on Mac - Apple Support

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By "circular Mac Pro", I take it that you mean the Mac Pro (Late 2013) desktop, a.k.a. "trash can", which has six Thunderbolt 2 ports. That would imply that you are using the Apple Thunderbolt 3-to-2 adapter to plug in your Thunderbolt drive.

The adapter only translates Thunderbolt. It does not carry power from one side to another. If that drive is a bus-powered one and it is plugged into the adapter (not into a Thunderbolt dock that has its own power supply), that configuration will not work because the drive will not be receiving any power.

If that is the case, you need to

• Plug the drive into its own power supply (if it is one that can optionally take external power),
• Get a powered Thunderbolt 3/4 dock and put it between the adapter and the bus-powered drive, or
• Get a USB hard drive or SSD. (This could be one with a USB-C plug, if you used the right adapter)