It was narrated that Abul-Tufail said: I said to Ibn 'Abbas (رضي الله عنه);

Tell me about riding between as-Safa and al-Marwah, for your people are saying that it is Sunnah. He said: They are telling the truth and they are lying. I said: How are they telling the truth and lying? He said: The Messenger of Allah (ﷺ) came to Makkah, and they came out; even the young women in seclusion came out. No one would be pushed away from the Messenger of Allah (ﷺ), so the Messenger of Allah (ﷺ) went between (as-Safa and al-Marwah - i.e., sa'y) riding. If he had dismounted, walking would have been dearer to him.