Animal sacrifice (‘Qurbani‘) is primarily part of the rituals of Hajj and is an optional ritual. The rituals of Hajj are reduced in the form of Umra with an optional animal sacrifice part of rituals of Umra as well. The same animal sacrifice is then extended for Muslims all around the globe who cannot afford Hajj to observe animal sacrifice in their places of residence, as a ‘Nafl ibadah‘ of course. Therefore, the primary Qurbani is that of Hajj which for people who cannot afford it, are allowed to conduct it at the places of their residence. The person who is going for Hajj will perform Qurbani along with the rituals of Hajj.
This animal sacrifice on 10th of Zulhajja is done by both the person performing Hajj and the person residing at home, but the only difference is that for the former, it is done as part of rituals of Hajj (with no Eid Salah) while in case of the latter, it is done after Eid Salah. This is how the Sunnah was established by the Prophet (sws).