Antimagic field only suppresses areas of effect while they overlap with the field; they function as normal otherwise
The antimagic field spell description answers your question:
A 10-foot-radius invisible sphere of antimagic surrounds you. This area is divorced from the magical energy that suffuses the multiverse. Within the sphere, spells can't be cast, summoned creatures disappear, and even magic items become mundane. Until the spell ends, the sphere moves with you, centered on you.
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
Areas of Magic. The area of another spell or magical effect, such as fireball, can't extend into the sphere. If the sphere overlaps an area of magic, the part of the area that is covered by the sphere is suppressed. For example, the flames created by a wall of fire are suppressed within the sphere, creating a gap in the wall if the overlap is large enough.
As the phrasing of the sentence I've bolded suggests, the area of effect of other spells and magical effects is suppressed only while they overlap with the antimagic field.
Those parts of an AoE outside the area of antimagic field are not suppressed (though I'm not sure how it works if the point of origin of a spell like darkness is within the area of antimagic field). Once either the other AoE is moved or you move (since antimagic field is centered on you) such that there is no overlap, there is no AoE in the field to suppress.
This interpretation is further supported by the spell's description of its interaction with creatures/objects summoned or created by magic:
Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.
As you can see, antimagic field doesn't dispel/end the other spell; it simply "suppresses" its effects while the field overlaps the effect of the other spell (or magical effect).