Discovered so far:
In Generation 2, shininess is determined by IVs - specifically, Speed/Defense/Special must be equal to 10, and Attack must be 2/3/6/7/10/11/14/15. (Note that IVs only go up to 15 in the Game Boy games). Pokebank attemps to take this into account, but gets the role of the Attack and Defense IVs exactly backwards. Meaning, Pokemon that would be shiny in GSC lose their shininess, and other pokemon will gain shininess.
Pokemon will not be allowed to transfer if they have moves that can be learned in one Gen 1 game, but not another. For example, your friend can transfer his Charizard with Fly that he taught it in Yellow, but if he trades it to your copy of Red it cannot be transferred.
Gender when transferring is completely random, rather than based on the Attack IV like in gen 2. Considering the above (failed attempt at) shiny compatibility, they obviously are trying to keep everything consistent with GSC, but random genders mean that - once GSC is released for VC - a Pokemon traded from gen 2 to gen 1 to gen 7 will lose its gender.
There's also a few other… oddities, shall we say, with how the conversion is done. Ones that aren't outright bugs like the above, but are surprising.
Nature is generated from - of all things - the Pokemon's EXP. This means that 1) catching the same Pokemon at the same level and immediately transferring will always get you the same nature, and 2) it is simple to choose the nature you want by only transferring when the experience is right.
EVs are simply set to zero, the Pokemon's previous stat exp is ignored entirely. This is actually the only thing on the list that makes sense, since the two systems are fundamentally incompatible.
The original IVs are also ignored entirely. After being used for the (incorrect) shiny calculation mentioned above, they are simply regenerated from scratch. All Pokemon will automatically have 3 perfect IVs, except for Mew which will automatically have 5.
Kudos to SciresM for discovering most of this.
Zai Redwinters also stated in a comment on my latest video that legal moves are based on a Yellow version whitelist. This goes against the claim that Charizard knowing Fly cannot be transferred over from a Red/Blue, but they give the example of Mewtwo knowing Pay Day (in all Generation I games except for Yellow) being unable to be transferred over in a certain version (or possibly both versions).