Obsoleting a building (or wonder) has a few effects. It effectively makes the building 'worthless'. In short: The building becomes useless, you can't make more, and the only thing that stays is it's culture.
In your example: If you research the advanced flight technology, then the stable becomes obsolete. You can no longer build any more, and the existing stables no longer provide XP to mounted units. It is an unusual thing to obsolete... advanced flight unlocks gunships, which replace calvalry, which means that you *can't* build any more mounted units anyway. But if you were lucky, and got the event that provides +1 food to all stables, all cities with stables in it will also lose that food too)
The nuances of obsoletion (for the obsessive) are:
* You can no longer build the building
* Existing buildings no longer provide food, production, commerce, beakers, gold, xp, great person points, happiness or healthiness. Both percentage (eg. the monasteries research bonus) and flat (eg. The Khmer's aqueducts +1 food bonus) are removed. Unhealthiness and unhappyness penalties are also removed.
* All wonder effects are removed (Eg. +50% worker speed)
* Free buildings made by the wonder are removed.
* In general, if it is not culture, then it is removed.
* AFAIK: If a wonder which unlocks civics is made obsolete, then you can no longer adopt those civics. However, if you were in one of the unlocked civics, then you stay there. Thankfully, both Swedigon Paya and the Pyramids never obsolete.
* Culture from the building is retained. So the monument's culture is retained.
* If a unit requires a building (Eg. missionaries for monasteries), then the unit can still be built. Thanks Sincro
* If you build a monument, and then build stonehedge, then when stonehedge is obsoleted, then all cities which didn't have a monument before building stonehedge, lose thier monuments. However, the monuments which were built before stonehedge stay, and thier culture boost stays. [The happiness boost for Charismatic Leaders is removed, however]
