BigCommerce has an API call limit depending on which plan you're using. Here is the API call limit for Plans:
Trial: 20000 per hour
Standard: 20000 per hour
Plus: 20000 per hour
Pro: 60000per hour
Enterprise: limitless
If you have reached the API call limit the migration process still going to continue, however, it's going to take some time. To surpass the limits you can consider upgrading your plan. For more information, you can check this link