There are two main things to consider in this
1) Games that normally did online play or LAN play*. This does include handhelds that used link cables.
2) Games that played on one screen with one system.
*you mentioned Mario Kart and thus Mario Kart double dash is in the mix here as it originally had a LAN mode that many stuffed over a VPN, can even have 8 players which makes things more interesting than some imagine.
For 1) you have to emulate the connection somewhere in the mix. Some things can do this with
For 2) you just need the ability to stream video to them and take commands back in.
Depending upon the game. Get them to join your computer via teamviewer or some similar VNC, though if they insist on using controllers their end then that can make life tricky (still ways, usually converting joypad to keys and sending those). Some will do 3d properly these days too and latency can be acceptable, though I would still prefer a turn based game if you can as latency is no fun.
http://www.kaillera.com/ is the classic choice for many, though long outdated and without some of the fun NAT traversal stuff more modern emulators might have built in.
https://emulation.fandom.com/wiki/Netplay also worth reading here, though will involve you setting things up**.
I have heard some people say good things about
https://www.fightcade.com/ but have not properly looked at it myself, and I don't think it will have any Mario Party support on any of its choices.
**nothing you can't really do over a teamviewer connection or similar.