If I buy an Iphone 13 pro from USA along with apple care warranty. Does the warranty work in other countries, especially India

Please refer to ---> See AppleCare+ Terms and Conditions for complete product details.

Please read this para...(Extracted from the above link)

"Service will be limited to the options available in the country where you request service. Service options, parts availability, and response times may vary. If service is not available for the Covered Equipment in a country that is not the country of purchase, you may be responsible for shipping and handling charges to facilitate service in a country where service is available. If you seek service in a country that is not the country of purchase, you must comply with all applicable import and export laws and regulations and be responsible for all custom duties, V.A.T. or GST, and other associated sales taxes and charges. For international service,

Apple may repair or exchange products and parts with comparable products and parts that comply with local standards."

This discussion thread may give you an answer --> Apple care plus International warranty.

IMPORTANT RESTRICTION FOR iPHONE, iPAD, APPLE TV AND HOMEPOD SERVICE.

Apple may restrict warranty service for iPhone, iPad, Apple TV and HomePod to the country where Apple or its Authorized Distributors originally sold the device.

You will surely get services from Apple Authorised Service Centres but may not be free.