I would suggest you to think in your use cases and then decide which related entities are you going to use for search purposes. Once decided, you can do two things:
Option A, use a nested field. Is simpler but, if you update a child (related) entity, the Elasticsearch index will not be updated. Yo must ensure that this happens within your code. For example, each time the child entity gets updated, you set a timestamp in the parent entotity, so FosElasticaBundle catch's that chahge and runs the entity serializer so the child entity becomes serialized.
Option B, use a child parent relation in Elasticsearch. In this case FosElasticaBundle will track your entity changes correctly for each Doctrine entity, so you don't need to manage it in your code.
In both cases, you many to many relationship becomes in two one to many relations. In option A this is done by the serializers that embeds the child document in the parent one. In the second case it is tranlated to a parent child relationship.