You can choose if you want to have fulfillment statuses sync between Katana and BigCommerce two-way:
If a Delivery status for a SO is changed to Partially Delivered in Katana, the fulfillment status in BigCommerce will not change.
If a Delivery status for a SO is changed to Delivered in Katana, the fulfillment status in BigCommerce will change to "Completed".
If a fulfillment status for a SO is changed to "Completed" in BigCommerce, the Delivery status in Katana will change to Delivered.
If fulfillment is initiated from BigCommerce and the SO has batch trackable items, batches are automatically assigned to your order using the FIFO (first in first out) method. See more here.