Once the appointed place had been reached, the messenger would tie a thread of scarlet wool around the horns of the scapegoat (m. Yom. 6.6; cf. Barn. 7.8). A second crimson thread was tied to the door of the temple sanctuary. Then 'the messenger pushed the [scape]goat from behind; and it went rolling down, and before it had reached half the way down the hill the [scape]goat had broken into pieces' (m. Yom. 6.6). According to tradition, 'when the he-goat reached the wilderness [and had been pushed down the cliff] the thread [tied to the sanctuary door] turned white; for it is written, “Though your sins be as scarlet, they shall be as white as snow (Isa 1.18)”' (m. Yom. 6.8). The Targum Pseudo Jonathan theologises the death of the scapegoat by making it a supernatural event: 'And the goat will go up on the mountains of Bethhadurey, and a tempest wind from the presence of the Lord will carry him away, and he will die.'
The custom of killing the scapegoat may have arisen because the scapegoat did, on occasion, find its way back to Jerusalem, perhaps in search of food and water. Such an event would be considered dangerous, threatening to undo all that the Day of Atonement had accomplished. This scenario could be prevented by assuring killing the goat after its expulsion. [B Hudson McLean The Cursed Christ p. 82 - 83]
Yet as the narrative in the Epistle to the Hebrews shows, the 'historical details' coincide with the expectation created by the priestly narratives of the Pentateuch. The bodies of the animals sacrificed as sin offerings were burned outside the camp (Leviticus 16:27). Of course it can - as has been argued - that Hebrews here is merely 're-interpreting' the historical details of Jesus's death in light of those priestly narratives in Leviticus - i.e. the crucifixion 'really' took place outside the city walls and then the 'sin offering' spin was added. However this avoids the reality that the entire Passion narrative has this priestly narrative.
For instance the abuse that is described in the gospel as being heaped upon Jesus 'the divine scapegoat' is derived from the ritual described above. So too the 'stations of the Cross' - which while not explicitly part of the 'gospel narrative' form the earliest ritual re-enactment of the narrative. Indeed the Latin Passion plays still preserve what we may identify as 'remembrances' of parts of the ur-gospel - i.e. parts of the heretical text which were excised when the orthodox texts were constructed (i.e. Berenice wiping the face of Jesus etc). This follows a pattern which we will examine in more detail here - namely that the earlier heretical gospels actually play up the scapegoat ritual elements meaning - quite evidently - that the orthodox texts rather than 'preserving' historical details only made the existing narrative 'seem' more historical by removing implausible details which were deemed to be incompatible with history.