As a follow-on note, mongosync / live migration requires the migration to be cut over for consistent reads on the destination cluster (see mongosync docs and Live Migrate docs for more details). If building the Atlas Search index mid-sync, there may be some lag time between when the migration is completed and the Atlas Search Index being consistent, as Atlas Search Indexes are also eventually consistent.
Echoing Seth’s recommendation above, Atlas Search Indexes should be built on the destination after the migration is cut over. Note also that writes should not be accepted on the destination cluster until after Atlas/mongosync indicates that it is safe to do so.

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