The Messenger of Allah ﷺ, when performing tawaf during Hajj or Umra upon his first arrival, would walk briskly during the first three circuits and walk normally during the remaining four, then he would pray two units of prayer.
حدثنا قتيبة بن سعيد حدثنا يعقوب عن موسى بن عقبة عن نافع
أن رسول الله ﷺ كان إذا طاف في الحج والعمرة أول ما يقدم فإنه يسعى ثلاثة أطواف ويمشي أربعا ثم يصلي سجدتين
Its citation, Bukhari narrated it in the Book of Hajj, chapter of one who performs tawaf around the Kaaba upon arriving in Makkah before etc, hadith 1616, and Muslim narrated it in the Book of Hajj, chapter on the recommendation of running during tawaf in Umrah, hadith 1261, from the hadith of Musa bin Uqba through the same chain.