Begin by registering for an API key on the New York Times Developer Network. This key will allow you to make requests to their API endpoints. Familiarize yourself with the available data and the structure of the JSON responses to understand how to extract the required information.

Prepare your MySQL database by creating a new database and defining the necessary tables to store the data. Ensure the table schema matches the structure of the data you plan to extract from the New York Times API. For example, create tables to store articles with fields like `id`, `title`, `abstract`, `url`, `published_date`, etc.

Write a Python script to make HTTP GET requests to the New York Times API endpoints. Use libraries like `requests` to handle these HTTP requests. Parse the JSON response using Python’s built-in `json` module to extract relevant data fields. Make sure to handle different types of data and any potential HTTP errors or exceptions.

Once you have extracted the data, transform it into a format that is compatible with your MySQL table schema. This may involve cleaning the data, converting data types, and ensuring that all required fields are populated. Prepare the data as a list of tuples or a dictionary that can be easily inserted into MySQL.

Use a library like `mysql-connector-python` to establish a connection to your MySQL database. Make sure to securely handle your database credentials and establish the connection using proper error handling to manage connection issues or authentication failures.

With the database connection established, write SQL `INSERT` statements to insert the transformed data into your MySQL tables. Use Python's cursor object to execute these SQL statements. Make sure to handle transactions properly, using commit and rollback as necessary to ensure data integrity.

To keep your MySQL database updated with the latest data from the New York Times, schedule your Python script to run at regular intervals using a tool like `cron` on Unix-based systems or Task Scheduler on Windows. This will automate the process of fetching and inserting new data, ensuring that your database remains current.

By following these steps, you can move data from the New York Times to a MySQL destination effectively without needing third-party connectors or integrations.