How should Regression Testing be carried out in an Agile Project

The simple answer to your question is “No - it doesn’t differ from a scrum team process.” Regression testing should be automated into the Agile Project before a sprint finishes.

The entire idea of an Agile Project Management is that the project can shift and evolve through the sprint while debugging certain issues that arise from adding additional elements and features. This automation ensures the product or task isn’t broken at the end of the sprint, but is ready to be handed on to the next sprint or is a finished final product.