I don't know much about the 3ds specifics.
As a basic premise then network packets can be captured and sent over private networks to be replicated far away. This is basic VPN and has been done for years (see xlink kai on the original xbox and things going back further still on PC).
The main problem with that tends to be network code not built to handle it
https://www.gamedeveloper.com/desig...r-what-i-learned-coding-x-wing-vs-tie-fighter is a very old article but as relevant as ever because... physics is not exactly going to change any time soon. This sort of thing is more likely to be accepting of a dropped packet than a wired connection but proper lag handling is still extra code over baseline transmission methods so it will not tend to be there.
Some devices, like unmodded xbox 360 hosts (unmodded boxes can join modded hosts), have a ping limit as well that is set around "maybe on the same ISP in the same town/cabinet".
The "can be captured" part however tends to be where the fun arises (broadcast less of one as you can presumably dummy up a packet).
The original DS' communication protocol, usually dubbed nifi, uses almost wifi packets but chops the header off. Most likely this is done for patent reasons; wifi is probably still patented in some parts of the world, and enforced (Australian CSIRO probably being the most noted in this), and almost certainly still would have been active when the 3ds was new. See also why DS wifi games carry firmware updates to add wifi functionality to those earlier models that might not have shipped with it, and why the original xbox did not play DVDs unless you got a remote control (and presumably why modern windows dropped DVD playback as stock).
Anyway this real oddity for packets means even most fun open source driver cards that Linux peeps or password cracking peeps recommend are still going to struggle and you start heading into either the rare ones can be cajoled into it (there are a few with various highly programmed internals), software defined radio, maybe cannibalising a (3)ds both sides with probably enough wires to look like a horror film/hardcore hacking scene to act as your radio or more hardcore custom electronics and test gear (you could probably do it with a lot of modern spectrum analysers, though not all and it is not a cheap hobby) which both sides of the equation will need. Not likely going to be a "oh buy an extra wifi card/wire your laptop to the router and bridge the connections and install a copy of wireshark*".
It is also why on the DS many will probably look at doing full wifi replacement efforts instead (most wifi setups for the DS being glorified handshake and multicast), even if the wifi options in games were not usually more feature rich.
I don't know if the 3ds has something similar to this, would not surprise me in the slightest and if you are within spitting distance for it then the need for full bore network comms/full bore OSI model implementation is technically diminished (nobody is likely to use that as a real reason though).
*modern consumer/non server Windows (post XP SP2) is also going to be tricky here as they stopped all the fun network things in the name of security (technically correct but nobody buys it) so you either get to have fun with ARP poisoning (and following the events of the pokemon scanner stuff on the 3ds**... I could do with a giggle again I suppose) or just go Linux live CD on top of that.
**sounds like you are at least baseline versed in network stuff so I will the basic overview and leave you to go finding the threads around here somewhere on it. Nintendo had their pokemon battles broadcast the opponents selections (full stats and all so was useful for determining hidden numbers prior to the 3ds being widely hacked as well) in nice unencrypted cleartext before the opponent had necessarily selected theirs and with more than enough time for you to make a counter selection. I was manning the reports section of the sister download site at the time and the amount of whining reports that came in claiming it was bad, illegal, wrong, a hacking tool (funny that on a hacking site) was amazing and keeps me amused to this day.
I suppose technically you also have the software option of modding each game hard enough that it uses real wifi protocols you can fire through a tunnel/VPN... there are a lot more projects I would expect before those become widely seen.
like a 250 dollar kit back when DDR4 was Intel only
