Posting for others searching for this problem. I recently installed Debian Buster on a new setup with an MCI motherboard, which has an onboard Realtek RTL8111/8168/8411 Ethernet Controller (use "lspci" to check that the controller is recognized).
During install, Debian installed an Ethernet driver called r8169, which is very unstable (use "lsmod" to see which driver is loaded). r8169 works for a while, then completely stops working after a few minutes.
The solution is to download the "official" driver (called r8168) from the Realtek site, then untar it and run an included script to build/install it. Here is an outline:
https://www.unixblogger.com/how-to-g...updated-guide/
Note that the download link on the page above is broken - you'll need to search through the Realtek site for the driver download page. Also, if you're on a new install, you'll need to install the kernel headers and programs necessary to build the driver, like the "build-essential" package on Debian.