Begin by accessing the Zendesk Chat REST API. You will need to authenticate using your Zendesk credentials. Typically, Zendesk provides an API token that you can use to authenticate. You can generate this token in your Zendesk account under the API settings. Ensure you have the appropriate permissions to access the chat data.

Use the Zendesk Chat API to fetch the data you need. You can perform HTTP GET requests to endpoints such as `/api/v2/chats` to retrieve chat data. Be sure to handle pagination if there are large volumes of data by checking the response for pagination links or using parameters like `page` and `per_page`.

Once you've retrieved the data, it will likely be in JSON format. Use a programming language like Python, JavaScript, or PHP to parse this JSON data. Libraries like `json` in Python or `JSON.parse()` in JavaScript can be used to convert the JSON string into a data structure you can manipulate, such as a list or dictionary.

Set up your MySQL database to receive the data. Create a database and the necessary tables that correspond to the structure of the data you have retrieved. Define the appropriate data types for each column to ensure that the data is stored correctly. For example, you might have tables for chats, users, and messages.

Establish a connection to your MySQL database using a suitable client library for your programming language. For example, you can use `mysql-connector-python` for Python, `mysql` module for Node.js, or `mysqli` for PHP. Ensure you have the correct credentials and permissions to insert data into the database.

With a connection established and data parsed, iterate over your data structure and insert records into your MySQL database. Construct SQL `INSERT` statements or use prepared statements to efficiently insert multiple records. Be mindful of data integrity and error handling, especially if you're inserting large volumes of data or performing batch inserts.

To keep your MySQL database updated with the latest data from Zendesk Chat, automate this process. You can write a script that regularly fetches new data and inserts it into the database. Use a cron job (on Linux) or Task Scheduler (on Windows) to schedule this script to run at regular intervals, such as every hour or day, depending on your needs.

By following these steps, you can effectively transfer data from Zendesk Chat to a MySQL destination without relying on third-party connectors or integrations.