Women, the elderly, children, sick pilgrims, and those who are weak or have difficulty with crowds are permitted to leave Muzdalifah after midnight (or after the moon sets) according to the consensus of scholars from all four schools. This is a confirmed Sunnah concession granted by the Prophet (peace be upon him) himself. He sent the weak members of his family, including his wife Sawdah and others, ahead from Muzdalifah to Mina during the latter part of the night to avoid the rush of the main body of pilgrims departing at dawn.
Sheikh Ibn Baz explained that the concession is available to all who have a genuine reason for early departure, including women who want to avoid the extreme crowding, elderly pilgrims, those with small children, and the sick. These pilgrims may proceed directly to Mina and stone Jamrat al-Aqabah before dawn, which is permissible for them based on this concession. Strong, able-bodied men should remain at Muzdalifah until after Fajr prayer, following the Prophet's own practice.
Sheikh Ibn Uthaymeen specified that the earliest permissible time for departure is the latter half of the night (after midnight), based on the narration that the Prophet sent them ahead 'at the end of the night' and the moon setting. He cautioned against departing too early — shortly after Maghrib and Isha prayers — as this would mean the pilgrim barely stayed at Muzdalifah at all. Those who take this concession should still pray Maghrib and Isha at Muzdalifah before departing.