The Silmarillion doesn't really specify in what manner he was casted out.
I have a couple scenarios that could be how:
He was literally thrown by Tulkas and Orome.
He was dragged by the collar and the Valar blasted him with their combined energy and he flew backwards into the void.
Does Tolkien give any more info than what is given about just him being cast out beyond the Walls of the World into the Void?
The war was successful, and ruin was limited to the small (if beautiful) region of Beleriand. Morgoth was thus actually made captive in physical form,9 and in that form taken as a mere criminal to Aman and delivered to Namo Mandos as judge — and executioner. He was judged, and eventually taken out of the Blessed Realm and executed: that is killed like one of the Incarnates. [...] When that body was destroyed he was weak and utterly 'houseless', and for that time at a loss and 'unanchored' as it were. We read that he was then thrust out into the Void.10 That should mean that he was put outside Time and Space, outside Ea altogether; but if that were so this would imply a direct intervention of Eru (with or without supplication of the Valar). It may however refer inaccurately* to the extrusion or flight of his spirit from Arda.
(HoME X)
But Morgoth himself the Gods thrust through the Door of Night into the Timeless Void, beyond the Walls of the World; and a guard is set for ever on that door, and Earendel keeps watch upon the ramparts of the sky.
(HoME 5)
That's all we know beyond what you already read.
He was delivered to Mandos in chains, Mandos judged and executed him, and the Valar thrust him into the Void. "Thrust" sounds physical, so presumably the Valar exerted their will "physically" somehow on the disembodied spirit of Morgoth, or it may have been that Eru just willed it so.
