This is achieved by making sure that the custom field values are already assigned to the respective list item (ie item, customer, vendor) prior to importing the transaction, see more on importing custom field lists, then when the transaction is imported the Custom field value will appear by default alongside the item, customer or vendor. This is called automatic propagation.
If you want to see this in action create an invoice in QuickBooks then add an item that you know has custom fields values assigned to it and you will see the custom fields and respective values automatically appear (make sure you are displaying the custom fields on the template). The same automatic propagation happens when Axis imports transactions.
Error Custom Field Values data format errors - Custom field data can be set to a specific format like Text, Numbers only, Dates, Multiple-choice list, Telephone formats. If you import data to the custom field that is not in the correct data format then you will get an error. You can check the required format for the custom data field by editing the field definitions in QuickBooks
Error - The specified custom field cannot be used by the list or transaction element - The error is due to the custom field that you are importing is not associated with the Name or Item you are importing it against. So for example you have a custom field for a customer but the custom field is only associated with an item. You an resolve this by setting up the custom field to be associated to the Name as well as the Item.