It appears that dispel magic has no effect
Living spells are not affected by dispel magic. Whatever magical effect is sustaining the sentience and energy of the living spells, it must not be a spell. Remember that dispel magic typically only works on spells, unless a specific effect is listed as being affected (emphasis mine):
Any spell of 3rd level or lower on the target ends.
That being said, non-spell effects can still be affected by dispel magic in specific cases such as the Animated Objects in the Monster Manual:
Antimagic Susceptibility. The object is incapacitated while in the area of an antimagic field. If targeted by dispel magic, the sword must succeed on a Constitution saving throw against the caster’s spell save DC or fall unconscious for 1 minute.
While the magical effect (the animation of the object) is clearly not entirely ended like a spell effect would be when hit by dispel magic, the antimagic does have some effect.
However, no such effect exists for living spells. This may be connected to whatever is causing “its magical energy (to endure) indefinitely.”
Antimagic field, however, will have an effect on the living spells.
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.
Living spells are creatures, so since they were created by magic, they would be affected by the above portion of the antimagic field spell.