Nafi' narrates from Ibn Umar (may Allah be pleased with them both) that the Messenger of Allah (peace and blessings be upon him) used to go out walking on the day of Eid al-Fitr and the day of Eid al-Adha, and a spear would be carried in front of him; then at the time of the prayer it would be planted in front of him so that he could take it as a sutrah (screen). This was before buildings were built at the Eid prayer ground. And he (peace and blessings be upon him) did the same at Arafah.
His word "walking" is gharib (unusual); I have not recorded it from the hadith of Ibn Umar except through this chain, and it is not strong. As for the rest of its wording, it is well known.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب صلاة العيدين / 6144
In the hadith of Sayyiduna Sa'd al-Qaraz (may Allah be pleased with him) it is stated that the Prophet (peace and blessings be upon him) used to go and return on foot (for the two Eids).
Hadith Referenceالسنن الكبرى للبيهقي / كتاب صلاة العيدين / 6145
Ali bin Abi Talib (may Allah be pleased with him) said: "Coming to the Eid prayer ground on foot is the Sunnah." Abu Dawud (may Allah have mercy on him) added these words in his narration: "When you return, you may ride."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب صلاة العيدين / 6147