It is narrated from Ibn Umar (may Allah be pleased with them both) that when the Messenger of Allah (peace and blessings be upon him) performed Tawaf during Hajj and Umrah, he would run during the first three circuits and walk during the remaining four, then he would pray two rak'ahs, and then perform the Sa'i between Safa and Marwah.
Hadith Referenceمسند الإمام الشافعي / كتاب الحج / 956
Hadith Takhrijأخرجه البخاري، الحج، باب استلام الحجر الأسود حين يقدم مكة .... الخ (1603) ومسلم، الحج، باب استحباب الرمل في الطواف في العمرة..... الخ (1266)۔
It is narrated from Ata (may Allah have mercy on him) that the Messenger of Allah (ﷺ) completed three out of the seven circuits by walking briskly (performing ramal), and there was no walking at a normal pace in between them.
Hadith Referenceمسند الإمام الشافعي / كتاب الحج / 957
Hadith Takhrijإسناده ضعيف لإرساله والحديث صحيح انظر الحديث الذي سبق برقم (956) أخرجه البيهقي في معرفة السنن والآثار (2942)۔
It is narrated from Ibn Umar (may Allah be pleased with them both) that he would perform raml from the Black Stone to the Black Stone, then he would say: "This is how the Messenger of Allah (ﷺ) did."
Hadith Referenceمسند الإمام الشافعي / كتاب الحج / 958
Hadith Takhrijصحيح من غير هذا الطريق أخرجه مسلم، الحج، باب استحباب الرمل في الطواف في العمرة وفي الطواف الأول في الحج (1263)۔
It is narrated from Masruq (may Allah have mercy on him) that he saw Abdullah bin Mas'ud (may Allah be pleased with him) begin by touching the Black Stone, then perform three circuits (tawaf) on the right side with a brisk walk (raml), and four circuits with a normal walk. Then he came to the Station of Ibrahim (peace be upon him) and offered two rak'ahs of prayer behind it.
Hadith Referenceمسند الإمام الشافعي / كتاب الحج / 959
Hadith Takhrijإسناده صحيح أخرجه البيهقي: 5 / 83 - وفي معرفة السنن والآثار له (2944) وابن أبي شيبة (14897، 15036)۔
It is narrated from Ibn Abi Mulaykah that Umar bin Khattab (may Allah be pleased with him) touched the Black Stone in order to perform Tawaf, then said: "Now, for whom should we expose our shoulders, and to whom should we show (our strength), when Allah has made Islam dominant? By Allah! I will perform Sa'i (Tawaf) in the same manner as they (i.e., the Prophet ﷺ) did."
Hadith Referenceمسند الإمام الشافعي / كتاب الحج / 960
Hadith Takhrijأخرجه البخاري، بمعناه من غير هذا الطريق الحج، باب الرمل في الحج والعمرة (1605)۔