There are many factors to be considered :
- Distance to target (Travel Time) and Open Borders
- Terrain
- available (standing) army (number and type of units)
- neighbours
Depending on distance, type of roads and bridges (river crossings), terrain, Open Borders with neighbouring civs, the travel time to reach a target city can be long or the target even can be unreachable.
In Civ 6 you also always have to consider that while you deal with an emergency your own empire can become one, too, when neighbouring civs backstab you. In one game (with 30 civs) I got 4-6 DoWs while on an emergency. If your army is too small to defend your own cities and take the target city at the same time, you better not join the emergency.
Emergencies probably would work better when the game would automatically draft neighbouring civs together with the player into an alliance as active and passive participants (open borders), ensuring that there is no backstabbing among neighbours and the road is free.
In the medieval ages, the pope was kind of moderator to organize the crusades, in 20th century it was rather the British Empire and later the UN / US, etc.
Game mechanisms might be similar to world fair in Civ5.
The player should be able to initiate emergencies if an emergency is not triggered. (Maybe depending on political advances.)
Friendly / neutral relations and open borders would have to be ensured for a certain time after achieving the target, too, to ensure safe return of the troops.
Diplomatic relations between participants might improve by a successful emergency.
There should be rules for taking (neutral and target) cities during an emergency and a possible (auto-)release of cities at the end of the emergency or consequences for not releasing cities.
Examples :
https://en.wikipedia.org/wiki/Crusades
https://en.wikipedia.org/wiki/Fourth_Crusade
https://en.wikipedia.org/wiki/Sack_of_Constantinople_(1204)
https://en.wikipedia.org/wiki/Crimean_War
https://en.wikipedia.org/wiki/Anglo-Soviet_invasion_of_Iran
https://en.wikipedia.org/wiki/Aftermath_of_World_War_II
https://en.wikipedia.org/wiki/Korean_War
https://en.wikipedia.org/wiki/Gulf_War
If a player is going for domination victory (or the game thinks so, e.g. when controlling 50% of capitals or occupying large number of cities/population), this might lead to an emergency alliance of (all) other players against the leading player. (World War)
This would probably be a feature not liked by everybody, so there should be an option to deactivate emergencies.