You are able assign an invoice to a different customer as long as the invoice is saved and not processed. Once an invoice is processed it cannot be changed in any way. 

1. Make sure the invoice is saved.

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2. Select the actions' star form the top right-hand corner of the invoice screen. Note: if your invoice is not saved, the 'actions' star wont be there.

3. Select 'Change Customer'

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4. Search for new customer

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5. Confirm Customer

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This will then return you to the invoice with the select customer as the current invoice customer.

Below is a short video to take you through these steps.