Proxy Hac (hac al-badal) has specific conditions that must be fulfilled for it to be valid. All four schools agree on the core requirements, with some variation in details. The fundamental conditions are agreed upon by the scholars based on the hadith evidence.
Sheikh Ibn Baz outlined the conditions as follows: First, the proxy must be a Muslim of sound mind and must have reached puberty. Second, the proxy must have already performed their own farz Hac before performing Hac for someone else — this is based on the hadith of Shubrumah. Third, the person on whose behalf Hac is performed must be either deceased or permanently unable to perform Hac due to old age or incurable illness. Temporary illness or temporary financial difficulty does not justify proxy Hac. Fourth, the proxy must make the intention (niyyah) at the miqat specifically on behalf of the other person, saying 'Labbayk on behalf of [name].' Fifth, the proxy performs all the rites exactly as they would for their own Hac.
Sheikh Ibn Uthaymeen added that the proxy should be someone known for their religiousness and trustworthiness, as they are performing a sacred act of worship. He also clarified that the proxy earns reward for their good deed, while the full reward of Hac goes to the person on whose behalf it is performed. If the proxy commits a violation during Hac that requires a penalty, they bear the penalty themselves, not the person on whose behalf they are performing Hac.