Backing up folders to a local network

My family has a file backup system stored on a local network, which we mount to our computers when we are at home. This works perfectly fine for everybody else in my family, but as the only person in my family who needs to use my computer away from home on a regular basis, storing my files on a network that I can only access from our house isn't ideal for me. I can keep all my files in folders local to my computer and manually back them up by copy-pasting them into the folders that are mounted to the network, but that's thousands of folders to copy and paste on a regular basis.

I need something that will periodically (maybe every set amount of time, maybe every time I restart, etc.) sync specified local folders with versions of those folders mounted on the network without trying to replace files that haven't been changed. The network's folder still exists when I'm not connected to the network or the relevant folders aren't mounted on my computer, but it's empty, so it needs to be able to check in some way whether the folders inside of it are actually mounted before trying to sync. Ideally I would also be able to manually initiate the process if I need to.

I'm willing to install an application for this, run a script, etc. but am not quite techy enough to figure out whether any of the backup solutions I've found will actually work for this use case or not. I'm also willing to change my local setup slightly to accommodate things, but I don't have control over the setup of the local network.

I'm running Ubuntu 20.04. Any recommendations are welcome. (And if there's a better place to post this, I'll take that too. I'm not sure if this is the right place for this kind of question or not.)