I asked the Messenger of Allah ﷺ about game caught by a falcon, and he said, "Whatever it catches for you, eat it."
حدثنا نصر بن علي وهناد وأبو عمار قالوا حدثنا عيسى بن يونس عن مجالد عن الشعبي
سألت رسول الله ﷺ عن صيد البازي فقال ما أمسك عليك فكل
[Its chain is daif] Abu Dawud narrated it in Kitab al-Sayd, Chapter on Hunting, no. 2851, from the hadith of Mujalid, and it is daif. The hadith has supporting narrations in mawquf form found in Bayhaqi, 9/235, 238, and elsewhere.