Before starting, identify the data entities you want to move from Asana to Snowflake, such as tasks, projects, or teams. Determine the data fields required and how they map to Snowflake’s schema.

Create a personal access token in Asana to authenticate API requests. Go to 'My Profile Settings' in Asana, then 'Apps', and generate a new personal access token. Note this token as it will be used for API calls.

Use Asana's API to extract data. You can use a scripting language like Python to make HTTP GET requests to Asana’s API endpoints. For example, use the requests library to fetch data like this:
```python
import requests

token = 'your_personal_access_token'
headers = {'Authorization': f'Bearer {token}'}
response = requests.get('https://app.asana.com/api/1.0/projects', headers=headers)
data = response.json()
```
Ensure to handle pagination if your data exceeds Asana's response limits.

Convert the extracted data into a format suitable for Snowflake. Typically, this involves converting JSON data into CSV format. Use Python or similar to iterate over the data and write it to a CSV file:
```python
import csv

with open('asana_data.csv', 'w', newline='') as csvfile:
fieldnames = ['id', 'name', 'due_date', 'completed']
writer = csv.DictWriter(csvfile, fieldnames=fieldnames)
writer.writeheader()
for task in data['data']:
writer.writerow({'id': task['id'], 'name': task['name'],
'due_date': task.get('due_on'), 'completed': task['completed']})
```

Ensure your Snowflake environment is ready to receive the data. This involves creating the necessary database, schema, and table structures. Use SQL commands in Snowflake’s web interface or SnowSQL CLI:
```sql
CREATE DATABASE asana_db;
USE DATABASE asana_db;
CREATE SCHEMA asana_schema;
CREATE TABLE asana_tasks (
id STRING,
name STRING,
due_date DATE,
completed BOOLEAN
);
```

Use Snowflake’s COPY INTO command to load data from your local CSV file into Snowflake. First, upload the CSV to a stage (e.g., using the Snowflake web interface or a command-line tool):
```sql
PUT file://asana_data.csv @%asana_tasks;
COPY INTO asana_tasks FROM @%asana_tasks FILE_FORMAT = (TYPE = 'CSV' FIELD_OPTIONALLY_ENCLOSED_BY = '"');
```

After loading, verify the data in Snowflake to ensure it matches the source data from Asana. Run SELECT queries to check the data:
```sql
SELECT FROM asana_tasks;
```
Regularly update the data by scheduling the extract-transform-load (ETL) process using a cron job or a similar scheduling tool, ensuring that the data stays in sync.

By following these steps, you can move data from Asana to Snowflake without relying on third-party connectors or integrations.