Burnsville Police say their officers first encountered a suspected car thief Sunday driving a vehicle with stolen plates, which he crashed. BPD says they were notified this same individual allegedly went on to carjack a female at gunpoint later in the day ... and then led police on a wild chase, all while he fired at officers who were in hot pursuit -- so claim the cops.

Part of that incident was captured by a bystander ... who filmed the suspect jumping out of the vehicle, and proceeding to run toward a highway with cruisers hot on his tail. In the clip, it appears the man attempts to carjack yet another vehicle on the road.

The white SUV ended up swerving out of the way, and police units finally caught up to him. The video freezes at that point ... presumably, that's when he was shot. BPD says the suspect was shooting at cops and officers returned fire, and the suspect ended up dead.

The officers involved in the shooting were placed on administrative leave, and the Bureau of Criminal Apprehension is investigating ... standard operating procedure.

Now, here's where things get interesting ... a local reporter for FOX9 -- who was covering protests at the Governor's mansion in St. Paul Sunday -- posted video of a group of demonstrators taking a knee in silence, this after they'd apparently heard there was another officer-involved shooting. Not just that ... but she says they planned to head down there too.

Later, she followed-up ... tweeting, "Just an update on this, protestors said later they're heading back to Brooklyn Center because they need more information about what happened in Burnsville first." In other words ... it doesn't look like they had any real info from the jump -- kneeling for a guy cops say was going full GTA.