First, pull the desired Docker image from Docker Hub and run the container. You can achieve this with:
```bash
docker pull
docker run
```
This step ensures you have the environment and data you wish to transfer available on your local system.

Access the running container to extract the data. Use the command:
```bash
docker exec -it /bin/sh
```
Navigate to the directory containing the data and use commands like `cat`, `cp`, or other utilities to prepare the data for transfer. If the data is in a file, you can copy it to your local system using:
```bash
docker cp :/path/to/data /local/path
```

Once you have the data on your local system, you need to format it according to Firestore's requirements. Typically, this involves converting it into JSON format. Use a script or tool to serialize the data:
```python
import json

# Example dictionary
data = {"key": "value"}

# Convert to JSON
formatted_data = json.dumps(data)
```

To interact with Google Firestore, you need to authenticate your requests. Set up a service account in Google Cloud Console with Firestore access and download the JSON key file. Set the environment variable:
```bash
export GOOGLE_APPLICATION_CREDENTIALS="/path/to/your/service-account-file.json"
```

Use Python with the `requests` library to write a script that uploads the data to Firestore using its REST API. Example:
```python
import requests

# Firestore URL
url = "https://firestore.googleapis.com/v1/projects/YOUR_PROJECT_ID/databases/(default)/documents/YOUR_COLLECTION"

headers = {
"Content-Type": "application/json",
"Authorization": "Bearer " + "YOUR_ACCESS_TOKEN"
}

# Load JSON data
data = {"fields": {"key": {"stringValue": "value"}}}

# Send POST request
response = requests.post(url, headers=headers, json=data)

print(response.status_code, response.json())
```
Replace placeholders with your project details and access token.

Use Google OAuth 2.0 to get an access token for authorization. You can obtain this token via the `gcloud` command-line tool:
```bash
gcloud auth application-default print-access-token
```
Use the token in the authorization header of your HTTP requests.

Run your script to transfer the data. Verify the success of the operation by checking the HTTP response and inspecting the Firestore database via the console or another script that retrieves the data:
```python
import requests

url = "https://firestore.googleapis.com/v1/projects/YOUR_PROJECT_ID/databases/(default)/documents/YOUR_COLLECTION/YOUR_DOCUMENT_ID"
response = requests.get(url, headers=headers)

print(response.status_code, response.json())
```

This approach allows you to move data manually from a Docker container environment to Google Firestore, leveraging the REST API for a direct, hands-on transfer without third-party tools.