In most U.S. states, a citizen's arrest using proportionate non-deadly force is authorized when the citizen has witnessed the crime or has been asked by a law enforcement officer to assist in making an arrest. The U.S. Constitution is not violated by this authorization.
For example, in Colorado, citizens arrests (not made at the direction of a law enforcement officer) are authorized by Section 18-1-707(7) of the Colorado Revised Statutes, which states that:
A private person acting on his own account is justified in using
reasonable and appropriate physical force upon another person when and
to the extent that he reasonably believes it necessary to effect an
arrest, or to prevent the escape from custody of an arrested person
who has committed an offense in his presence; but he is justified in
using deadly physical force for the purpose only when he reasonably
believes it necessary to defend himself or a third person from what he
reasonably believes to be the use or imminent use of deadly physical
force.
Generally speaking it is illegal in the context of a citizen's arrest to use: disproportionate force, force applied to punish rather than detain someone, or deadly force, although there are times when deadly force is authorized for reasons similar to those that would apply for self-defense in the absence of a citizen's arrest. Deadly force is generally not authorized to protect tangible personal property or most forms of real property, but some places authorize the use of deadly force to protect a residence.
I've heard before that a thief have "right" to flee, and trying to
knock him out is not justified because you can only use violence to
protect your self, instead of your money.
While it is understandable that you might think this based upon U.S. Supreme Court cases like Tennessee v. Garner (1985), the law is actually more nuanced and that case held that:
when a law enforcement officer is pursuing a fleeing suspect, the
officer may not use deadly force to prevent escape unless "the officer
has probable cause to believe that the suspect poses a significant
threat of death or serious physical injury to the officer or others."
It was found that use of deadly force to prevent escape is an
unreasonable seizure under the Fourth Amendment, in the absence of
probable cause that the fleeing suspect posed a physical danger.
Thus, it is illegal to shoot to kill a fleeing thief or shoplifter, even though it is not illegal to restrain a shoplifter physically to prevent that shoplifter from fleeing the scene prior to the arrival of the police.