Reference / Source: This is in Ibn Sa'd's "al-Tabaqat al-Kubra" (3/115), narrated from Muhammad bin 'Umar al-Waqidi, from Abd Allah bin Ja'far, from Isma'il bin Muhammad bin Sa'd and Muhammad bin Abd Allah bin 'Amr; and (by another chain) from Muhammad bin 'Umar al-Waqidi, from Qudamah bin Musa, from 'A'ishah bint Qudamah. All of them mentioned the killing of Tulayb bin 'Umayr, its place, and his age at that time.

Details of the narration / Addendum: The mention of Tulayb's (may Allah be pleased with him) participation in the Battle of Badr is in Ibn Sa'd (3/114), narrated from al-Waqidi. Then Ibn Sa'd said: "Musa bin 'Uqbah, Muhammad bin Ishaq and Abu Ma'shar did not mention him among those who took part in Badr." So he said, although Musa bin 'Uqbah did mention him, on the authority of al-Zuhri, among those who took part in Badr, as Ibn Abi 'Asim recorded from him in "al-Ahad wa al-Mathani" (345).

Research note: Ibn 'Abd al-Barr said in "al-Isti'ab" (p. 362): "He took part in Badr according to the statement of Ibn Ishaq and al-Waqidi, but his name has dropped out in some narrations from Ibn Ishaq."

Reference / Source: Among those who counted Tulayb (bin 'Umayr) (may Allah be pleased with him) among the participants of the Battle of Badr are al-Kalbi (in "Jamharat Ansab al-'Arab"), Mus'ab bin Abd Allah al-Zubayri (in "Nasab Quraysh") and al-Zubayr bin Bakkar (see: Ibn 'Asakir's "Tarikh Dimashq" 25/142-143).
Important point: Hence the view of most historians and scholars is that Tulayb (may Allah be pleased with him) took part in the Battle of Badr.

Details of the narration: As for his being among those who emigrated to Abyssinia, Ibn Sa'd (3/114) says: Musa bin 'Uqbah, Muhammad bin Ishaq, Abu Ma'shar and Muhammad bin 'Umar (al-Waqidi) all mentioned him among those who made the second emigration to Abyssinia, and they all agreed on this.

Technical note: We say: Ibn Ishaq also mentioned him (see: Ibn Hisham's "al-Sirah al-Nabawiyyah" 1/324), and Musa bin 'Uqbah mentioned him on the authority of al-Zuhri (see: Abu Nu'aym's "Ma'rifat al-Sahabah" 3977).

Details of the narration: Musa bin 'Uqbah also mentioned from al-Zuhri (see: Abu Nu'aym's "al-Ma'rifah" 3977) that Tulayb (may Allah be pleased with him) was martyred at the Battle of "Ajnadayn". 'Urwah and Ibn Ishaq also said the same (see: Ibn 'Asakir 25/146-147).
Variation in the narration: However, Mus'ab al-Zubayri differed from them in "Nasab Quraysh" (p. 257) and said that he was martyred on the day of "al-Yarmuk". Al-Tabari also narrated the same (martyrdom at al-Yarmuk) in his "Tarikh" (3/402) from Sayf bin 'Umar, from Abu 'Uthman Yazid bin Usayd al-Ghassani and another shaykh named Khalid.