From the abstract for W. Burlette Carter, Can a Sitting President Be Federally Prosecuted? The Founders Answer, Howard Law Journal, 2019:
Exploring the history of impeachment and prosecution in (1) England and Great Britain, (2) colonial America, and (3) the states immediately after independence, and comparing these to the Founders’ Constitutional discussions, this article considers how the Founders would have answered that question, were it posed to them today. Deviating from most analyses of the problem, it argues that the Founders would have viewed the question as jurisdictional, involving a conflict between courts of law on the one hand, and the Congress–operating as a High Grand Jury (the House) and a High Court of Impeachment (the Senate)–on the other. They would have said that they gave to the Congress, constituted in its impeachment role, sole jurisdiction over removing a U.S. President for misbehavior. At the same time, they would have said that they gave courts of law concurrent power to hear cases involving crimes and misdemeanors so long as they do not involve removal. And so they would have answered our question with a question: Does the threatened action against the President risk removing the President, either directly or constructively? They would have believed that any order of a criminal authority affirmatively or negatively enjoining powers delegated to the President under the Constitution would shift those power to others and, therefore, would be a partial and impermissible removal attempt. Congress could insert itself into such proceedings to protect the Presidency (and, to remove or protect a President), but even if it does not, no federal court has jurisdiction to enforce such a removal order and, therefore a federal prosecutor has no power.
Consistent with British impeachment history, Founders would have viewed the proper issuance of Articles of Impeachment as a jurisdictional act that signaled Congress’ intent to actively intervene. As such, such Articles would automatically stay any contrary proceeding in or related to courts of law, including a federal investigation and the operation of a federal grand jury. Moreover, they would have said that a President has the power to fire his prosecutor, if he appointed him. And while a president can be impeached for obstructive behavior, they would have said that he cannot, after an impeachment, be prosecuted for statutory obstruction of justice, if the statutory prosecution is based on the exercise of powers delegated to him as President under the Constitution.
The Founders would have recognized that, before the formal issuance of Articles of Impeachment, courts of law have the power to stay their own proceedings against a President for good cause, just as English/British common law courts with concurrent jurisdiction always could. And they would have have accepted that courts of law can, in the first instance, decide evidentiary issues such as executive privilege for matters proceeding in their fora. Again, despite Parliamentary power over impeachment, common law courts had long done so in England and Great Britain, so long as they otherwise had jurisdiction.
This jurisdiction-focused answer from the Founders balances the interest in law enforcement in a given case with the larger interest of the nation in protecting the people’s investment in the Presidency. It allows prosecutors to investigate the behavior of a sitting President–up to a point. On the other hand, it not only authorizes but requires that prosecutors, sworn to uphold the Constitution, exercise prosecutorial discretion in determining whether or not to mount an investigation of a sitting President and how far to take it through the courts of law. And it requires that courts of law accept the limits of their jurisdiction when proposed injunctions or orders would threaten a removal.
The article sets forth the broad outlines of presidential removal doctrine and its jurisdictional ties. The focus is on the powers of federal prosecutors and federal courts in criminal proceedings vis a vis the President. It does not deal with the definition of high crimes and misdemeanors or many other related impeachment questions. While some of the arguments presented here might possibly apply to other types of proceedings against a President, including state prosecutions, this article focuses on federal prosecution.