A punishment is basically a harm to someone's person, liberty, dignity, or property imposed primarily for the purpose of harming someone, rather than to accomplish some non-punitive end such as apprehending a suspected criminal, securing compliance with a court order, exchanging truthful facts, or compensating someone for harm legally attributable to the person upon whom the act is imposed.
If a police officer shoots and kills a person while they are in the process of committing a crime, that is not a punishment, because the officer is not trying to punish someone for committing the crime, but to prevent the crime from continuing and/or to apprehend the suspect. If the use of force is justified by a law authorizing it under the circumstances, then it is legal and there are no civil or criminal consequences.
If the use of force is not justified by a law authorizing it under the circumstances it might be a crime, it might be a "tort" (i.e. a civil wrong for which you can sue someone, especially if it arises at common law), or it might be a civil rights violation. Law enforcement officers generally have broad absolute immunity from common law tort liability incurred the course of their duties in carrying out their jobs, and have "qualified immunity" from liability for civil rights violations which limits their liability to cases of intentional violations of clearly established constitutional rights.
In a case arising from an excessive use of force against someone who is not in custody, generally speaking, the constitutional right violated is the 4th Amendment right to be free of unreasonable seizures and to be free from seizures that that are not supported by probable cause.
Generally speaking, the law does not recognize an excessive use of force in connection with the criminal justice system as a "taking" for which there is a right to both due process and fair market value compensation (to somewhat oversimplify).
Generally speaking, an 8th Amendment analysis involving the use of force (as opposed to taking of money or property in the criminal justice system which are subject to the excessive fines clause of the Bill of Rights), begins, and the 4th Amendment seizure analysis ends, when someone is in custody.
A police officer who arrests someone and then beats them up or rapes them and then releases them, might be entering into 8th Amendment, rather than 4th Amendment territory, although the dividing lines are not always clear.