And to add on, specifically for Gen 1 <-> Gen 2 trading, the Catch Rate may need to change.
When trading from Gen 1 -> Gen 2, the Catch Rate data slot is usurped and used for Held Items.
That is how some mons caught in Gen 1 come with Held Items. The game just converts the value in that datapoint to an held item.
However, it is important to note some values in that slot don't correspond to a working item, so that game takes that into account and changes the value in that slot.
So if such a mon was then traded back from Gen 2 -> Gen 1, the modified value is kept.
TL;DR: For certain Gen 1 mons, if their catch rate is unchanged, it implies it was never traded to Gen 2.
Read more here: https://bulbapedia.bulbagarden.net/wiki/List_of_Pokémon_by_wild_held_item_(Generation_II)#Generation_I_held_items