Praise be to Allah.
It is not one of the conditions of Eid prayer being valid that all the people should pray in the same place, but it is better for them to pray in one place in the desert (i.e., outside the city), if that is possible. If it is too difficult for them all to pray in one place outside the city, such as the city being too spread out, then it is permissible for them to pray in two or more places outside the city, based on whatever is easiest for them.
If it is too difficult for them to pray out of doors because of rain etc., then they should pray in the mosque if it is big enough and that is not hard for them. Otherwise they may pray in a number of mosques, each group in the mosque in which it is easy for them to pray.
Secondly: in the event that there are a number of places for Eid prayers outside the city or in the mosques, it is permissible for one group to start the Eid prayer before another, or to end the prayer before another group, so long as the prayer of each group takes place between the time when the sun has risen to the height of a spear above the horizon and its passing the meridian, when the time for Zuhr begins, i.e., from the time when naafil prayer becomes permissible until the time when the sun is directly overhead in the sky, just before the time for Zuhr begins.
And Allaah knows best.