Citation pilot reported they were instructed to hold short of an intersecting runway but the pilot taxiing entered the runway as an aircraft taking off passed immediately in front of them.

Narrative

I was acting as Second in Command and Pilot Monitoring on Aircraft X; conducting a Part 135 flight from ZZZ to ZZZZ. The Pilot in Command and Pilot Flying made an uneventful landing on Runway XX at ZZZZ. ZZZZ Tower instructed us to exit Runway XX on taxiway 1 and hold short of Runway XY. I acknowledged and read back these instructions. The pilot flying initiated a moderate speed turn onto taxiway 1 and did not appear prepared to stop before the Runway XY hold short markings. I then reminded the pilot flying that we were instructed to hold short of Runway XY and pointed out the hold short line; however; he made no attempt to stop before the hold short line; and taxied past the Runway XY hold short markings. Meanwhile; another aircraft had been cleared for takeoff on Runway XY and was on the takeoff roll. After I raised my voice further and said 'STOP' the pilot flying with a slightly delayed reaction then abruptly stopped the aircraft before fully entering the runway. After the other aircraft on takeoff passed immediately in front of us; we were instructed to taxi to the ramp via taxiway 2; and shut the aircraft down without incident. As a result of the events of the day; while acting as Pilot Monitoring; I will take additional steps to be more assertive in similar situations and ensure I am able to take control of the aircraft to avoid situations like this.