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The process of transferring data obtained through StreamBuilder to swipeable cards sourced from Firestore in Flutter involves the following steps:

1. Create a StreamBuilder widget that listens to the data stream from Firestore.

2. Define your card layout and create a custom widget that displays the data for each card.

3. Use the StreamBuilder to pass the data to each card widget as it is received from Firestore.

4. Wrap the card widget in a Dismissible widget to enable swiping.

5. Implement the onDismissed callback to remove the dismissed card from Firestore.

6. Optionally, you can use an AnimatedList widget to provide a smoother transition when cards are added or removed from the list.

7. Update the list displayed in the UI as new data is added or removed from Firestore using the StreamBuilder.