Are entity increment id's migrated as-is from the source to the destination store?

No, during StagingPro migrations from the source store to the destination store, we have no control of the increment ID’s assigned by BigCommerce’s systems when the data hits the target store. Which is why, as an example, products must use SKU as the point of reference, not product ‘increment’ ID. Such ID’s are internal reference numbers auto-generated by BigCommerce maintains and which StagingPro cannot control. Product SKUs will get moved. Similarly when you move Orders new Order IDs will get assigned by BigCommerce and the orders will be moved to Pending State so that a user does not fulfil the orders from the destination store, inadvertently.