The latest developments - filmed on Monday - are a deadly new phase of the fragmented country's war.
The videos gave a glimpse into the brutal closing stages of a pitched battle for control of territory near the strait the Houthis had captured in September. The Houthi advance sent shockwaves through the oil market and pushed up petrol and diesel prices worldwide.
Fighting continued near Bab el-Mandeb on Tuesday as Saudi-backed government forces battled to consolidate Monday's counter-attack, besieging Houthi fighters at a military camp near the strait and pushing into hills to the east, Yemeni military sources said.
The videos, verified by Reuters and filmed from inside the armoured vehicles, showed them striking fighters with their vehicles. Some were armed and some were walking or running. However, others were unarmed and some were sitting, lying or crouching on the sandy shore and made no move to evade the oncoming vehicle as they were run over.
At one point, a vehicle hits a man, running and unarmed, and another crouching on the ground. It then loops back to drive over two men, both with their hands up and unarmed. Shooting can be heard during the video and at one point the vehicle's windscreen is cracked by a gunshot.
A Houthi statement decried the videos as proof of war crimes, saying they showed "the true nature of the Saudi regime and the forces it has mobilised from extremist factions".
Asil al-Saqladi, spokesperson for the Giants Brigade — the main Yemeni government force involved in the fighting at Bab el-Mandeb this week — said the force was conducting an investigation into the incident.
The Saudi-led coalition backing the government forces in Yemen did not immediately respond to a request for comment on the videos or the Houthi accusations of war crimes.
The Homeland Shield Forces, a significant formation in the Yemeni government's military coalition, did not comment directly on the videos but issued a directive on Tuesday urging troops to treat prisoners well and adhere to religious, ethical and legal standards.
Rights groups have previously accused the Houthis of war crimes, including sniper and shelling attacks on civilian targets, seizures of aid, abuse of prisoners and use of child soldiers. The group has denied war crimes.
Reuters has not been able to independently verify the extent of government gains and the Houthis denied they had lost ground, saying the government claims of advances were baseless.
Intense fighting meanwhile continued around Yemen's third-largest city of Taiz, where Houthi gains in a mountain district have in recent days cut off a main road to Aden, leaving government forces encircled.
Yemen's civil war pits the Houthis against an uneasy coalition of sometimes rival military forces under the internationally recognised government, backed by Saudi Arabia and its allies.
The renewed clashes have added a new front to the Iran-US conflict, with the US rebuffing Saudi requests for military help to reverse Houthi gains on the Red Sea coast near Bab el-Mandeb last month and end the group's missile and drone attacks on their territory.
The Giants Brigade is a southern military formation originally formed by students at ultra-conservative Sunni Muslim religious colleges.
That background gives both a religious and regional edge to their enmity with the Houthis, who originate in north Yemen and belong to the Zaydi school of Shi'ite Islam.
The Giants have themselves fractured this year, splitting between factions backed by Saudi Arabia and the United Arab Emirates, illustrating the internal divisions that have long beset the Yemeni government's efforts to fight the Houthis.
As well as further squeezing global trade routes after Iran's disruption of shipping in the Strait of Hormuz, the conflagration in Yemen has brought the conflict more directly to Saudi Arabia, a US ally.
