Apple - Does Apple Remote Desktop connect to computers over the internet?

Yes it can use iCloud back to my Mac to locate remote Macs behind a NAT router. Also, if your network allows inbound network connections then that also would work over the Internet.

And no, ARD is not a service where a client runs on each Mac to tunnel out of more arbitrary network connections as Team Viewer, Citrix/GoToMeeting and other solutions that are designed to have the computer that is "to be controlled" initiate contact with the administrator.

For example, setting up the clients over Bonjour will not work if you then try to access the clients over the Internet.

1. Select the scanner.

   

2. Select the Network Address scanner type, and enter the public/BTMM IP address of the computer.

   

   Yes, the client computer will need a "real Internet IP address" for this method to work, or the Back to my Mac address if enabled.

To find the Back to my Mac address, make sure that iCloud and Back to my Mac is enabled, then run the following command:

dns-sd -E

The final line shows the account number.

Use the following syntax for determining the address:

computername.number.members.btmm.icloud.com

^{You can find the computer name in System Preferences → Sharing.}

I would add that the method mentioned to use the back to my mac to resolve the ip address behind a nat router will only work if the host and client are logged into the same icloud account. See https://apple.stackexchange.com/a/53784 for a description.

I have been unable to get the ARD to resolve with two different icloud accounts, but when I logged into the same account on client and server, it worked fine for ARD and ssh (as shown in the link.)

I would submit that the accepted answer will not work for the original poster, because the OP needs to manage multiple employee's laptops who will not have the same icloud account as the IT pro.

Back To My Mac is no longer available in macOS Catalina.

I tried using Apple Remote Desktop and the iCloud/Back To My Mac solution proposed here to remotely connect to my Mac Pro from outside of our LAN whilst travelling.

Welp, Back To My Mac apparently got killed off in a previous version of macOS.

I ended up using Screens and setting up our modem/router to allow for secure, external connections. Everything worked well.

Too bad we have to use a third party for something like this now but them the breaks, I guess.