Once Time Machines were first invented and then the secret of how to make them was leaked to the criminal underworld, which then made and sold illegal time machines for profit, people started going back in time to prevent murders from happening. Mostly this was done with the best of intentions by family members who didn't want to lose their father, mother, child or other family member. But there were political attempts as well to change the timeline, such as when a unit of Ulster Loyalist terrorists went back in time to the Irish War of Independence to try and prevent police detectives from being murdered and ensure that it was the British who won that war.
The Time Patrol decided that to stop a murder of the past from happening would endanger the timeline itself. That sweet little boy or girl who was murdered might grow up to become a serial killer or worse, whilst preventing the murder of a famous historical figure could utterly change a timeline, perhaps very badly. When an illegal time machine is used, they have instruments that can show where it is, and the exact time and place it is going to. They can then use their own time machines to send the Murder Squad to the scene a few minutes ahead, dressed in the uniforms and carrying the weapons of law enforcement or the nearest equivalent of law enforcement of the time. Where possible they will arrest rather then kill, so the illegal time travellers can face trial and punishment in their own time, but they are willing and able to kill where they have to, to either defend themselves from attack or protect the timeline from being disturbed, taking the body or bodies away with them if possible. If it's not possible, at the very least they will remove any future tech found on the bodies.
The normal penalty for trying to prevent a murder by going back in time depends on the motives. If it is just to save a family member it can be as little as a certain number of years in prison, but in more serious cases where the motive was clearly political in nature it can be life imprisonment or even the death penalty.
Plot Hooks- The PCs can be members of the Murder Squad themselves or can be trying to bring back a fellow PC by going back in time to save him or her and come up against them.