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 is 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