Apple Remote Desktop over internet “without” port forwarding

Apple Remote Desktop will involve either having a VPN or open ports and static IPs. I think you will be much better off with LogMeIn.com's free version, which will require zero intraction from your family members and open ports are not necessary. You will need to know a user's credentials on their computer. If they require more privacy they can disable LogMeIn from the menu bar item, as it will always be on as long as they are connected to the Internet. I suggest adding a user to each computer with your name and one password—that will be one less thing for you to remember and if they change their password or don't remember it you can still help them.

I use both ARD and LogMeIn daily, but ARD is more than you need.

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.}