Moving affiliate plugins is mostly a data import. The part that goes wrong is the unpaid balance, because it is the one number an affiliate checks and the one number nobody notices is off until they complain.
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Import history, not just people
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An import that brings across affiliates but not their referrals resets every balance to zero. Bring the referrals and the payout history together, so what is owed is derived from the same records it was derived from before.
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Reconcile before you switch off
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Keep the old plugin installed and deactivated for a while. Pick your ten highest-earning affiliates and compare their balances line by line in both systems. If those ten agree, the rest almost certainly do.
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Tell the affiliates
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Say the migration is happening, say when, and say what to check. An affiliate who has been warned will report a discrepancy; an affiliate who has not will assume you shorted them.