It appears it is an unused feature. That is to say, no downloadable tests were ever released for either Pokemon X & Y, or Omega Ruby & Alpha Sapphire.
I remember not being able to download tests when I played, and I cannot find any mention online of specific downloadable tests, or any special events that may have featured them. Serebii has the following on the X/Y Battle Institute page:
There is also a download option which will allow for the download of special training scenarios through the Nintendo Network, and locally, but it is unknown whether or not this shall be utilised as a feature.
Serebii.Net - Pokemon X & Y - Battle Institute - Retrieved 21 October 2019
And a similar message (without the final sentence) is on the Omega Ruby & Alpha Sapphire page:
There is also a download option which will allow for the download of special training scenarios through the Nintendo Network, and locally..
Serebii.Net - Pokemon Omega Ruby Alpha Sapphire - Battle Institute - Retrieved 21 October 2019
The rest of the more common sources of info surrounding the battle institute either simply mention it as a feature without elaboration, or don't mention it at all, as is the case with Bulbapedia's entry on the Battle Institute. One thread on GameFAQs simply states that there are 'no download tests yet'.
Furthermore, there doesn't seem to be any rom-hacking/modding scene that has investigated the 'download test' functionality to either add tests or alter game data.
Suffice to say it's a bit of dead functionality that never got used.