The book by Robert Drew admittably covers the fall of the Mycenian, Cretan and Hittite cultures as well as those of the Palestine area, Ugarit etc. This was also the time of the invasion of Egypt by the Sea People.
All these Bronze Age Civilisations based their power on elite corps of charioteers, and only wealthy rulers could afford them. Tactically they operated as weapons platforms for the Asiatic compound bow, and battles were often inconclusive affairs with masses of chariots swirling around each in complex manoevers. Infantry as such had merely a walk-on part in support of the chariots. Mostly they were armed with blunt impact weapons rather than swords.
There are numerous references to the use of "runners" or skirmishers who would literally run along with the chariots to protect them from enemy skirmishers. When the chariots speeded up the runners would hang onto the horses.
All this was before the Assyrian Empire of the eighth century BC, the period when I imagine the program was focussed, but by that time the golden age of chariot warfare was well and truly over, together with the kingdoms relying on it.
Per Drew, what finished the chariot as a decisive weapon was the impact of masses of javelin-armed infantry, who targeted the horses rather than the bowmen. The vastly expensive chariot rigs, horses and crew could only be afforded by wealthy kingdoms, whilst there were thousands of mercenary, backwoods javelin-throwing soldiers available. Result..instant democratisation of warfare similar to the introduction of the long-bow and firearms in later periods.
These guys weren't interested in negotiated boundary adjustments and dynastic evolutions and brought down every civilisation in the Eastern Med within a couple of decades. The Dark Age lasted until the seventh century bc. It's from this period that the legends of the Trojan War come from.
Interestingly enough the Assyrian kingdom was one of the few to escape the impact of the new warfare due to their remoteness from the Med itself. The massed javelin-armed raiders seem largely to have been sea-borne.
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Wolf may come, sky may fall....