What legal authorities provide for DHS to police cities?

Ooops! The federal arson statute, 18 U.S. Code § 844, is being used by the US Attorney in Seattle to charge a woman from Tacoma who burned several Seattle police cars during the May riots in Seattle. She was charged under sections that make it a federal crime to burn property: a) belonging to agencies getting federal grants, or b) used in interstate or international commerce.

This is a very narrow exception to the general rule that the Feds have limited jurisdiction over local crime.

There are no parallel statutes making it a federal crime to destroy property belonging to agencies that get grants, or property used in interstate commerce, or to interfere with interstate commerce in general. For example, the Hobbs Act, 18 U.S. Code § 1951 makes it a crime to obstruct or interfere with interstate commerce, but only by means of robbery or extortion…

Here are the sections of the statutes she is charged with violating:

18 U.S. Code § 844(f)(1): Whoever maliciously damages or destroys, or attempts to damage or destroy, by means of fire or an explosive, any building, vehicle, or other personal or real property in whole or in part owned or possessed by, or leased to, the United States, or any department or agency thereof, or any institution or organization receiving Federal financial assistance, shall be imprisoned for not less than 5 years and not more than 20 years, fined under this title, or both.

18 U.S. Code § 844(l): Whoever maliciously damages or destroys, or attempts to damage or destroy, by means of fire or an explosive, any building, vehicle, or other real or personal property used in interstate or foreign commerce or in any activity affecting interstate or foreign commerce shall be imprisoned for not less than 5 years and not more than 20 years, fined under this title, or both; and if personal injury results to any person, including any public safety officer performing duties as a direct or proximate result of conduct prohibited by this subsection, shall be imprisoned for not less than 7 years and not more than 40 years, fined under this title, or both; and if death results to any person, including any public safety officer performing duties as a direct or proximate result of conduct prohibited by this subsection, shall also be subject to imprisonment for any term of years, or to the death penalty or to life imprisonment.

Proving a negative is impossible, but right now, the best answers to your questions are:

No, federal statutes do not give the Feds authority to police cities. Federal law gives the Feds power to protect federal property and enforce federal and Oregon law on Federal property.

No, the Feds have not cited any statutes authorizing them to do more than protect federal property.

Since you should not believe me, let me quote Professor Steven Vladeck of the University of Texas. He has impeccable credentials, he wrote about this topic on a well-known and respectedweb site, and his article is referred to by other lawyers writing about Portland.

Here is his answer to the question, **which federal laws are being enforced?"

This is a significant question, and the answer is hardly obvious. Unlike D.C., which is entirely federal territory (and where even “local” law is technically “federal”), Portland is principally subject to the sovereignty of the state of Oregon. There is some federal property in Portland, but not much. And federal law enforcement officers do not generally have the authority (under federal law, anyway) to enforce state laws on nonfederal property.

Some of Acting Secretary Wolf’s statements have referred to vandalism of “courthouses,” including the three federal courthouses in downtown Portland. And there are a handful of other federal buildings in the city.

But this is where the Portland situation starts to look very different from D.C. Protecting a federal courthouse from vandalism is an easy case for the use of federal law enforcement authorities. If, as media reports have suggested, federal officers are patrolling streets a significant distance from federal buildings (and arresting protestors who pose no imminent threat thereto), that would be far murkier.

The fact that he does not cite any federal law answers your question. He didn’t just “end up” not providing cites to federal law. In the article, he explains why he can't cite any relevant federal laws:

1. The Feds haven’t cited any statutory authority beyond protecting Federal buildings.

2. As he (implicitly) says in the cited section, he can’t think of any statutes they could be enforcing.

3. In order to enforce Oregon law off of federal property, Oregon law requires the Feds to do a bunch of stuff they haven’t done.

There may be other experts who have found federal statutes that give the Fed authority to "restore order" in Portland, but I have not found any.

In the rest of the article, Vladeck discusses what authority Feds have to detain or arrest people off of Federal property. (Answer: Not the sort it sounds like they are exercising).

In related article he looks at Federal plans to surveil people who might be threatening monuments, including monuments that are not on Federal property.

You've omitted a critical part of paragraph (b)(2)(C):

make arrests without a warrant for any offense against the United States committed in the presence of the officer or agent or for any felony cognizable under the laws of the United States if the officer or agent has reasonable grounds to believe that the person to be arrested has committed or is committing a felony

So if the crime in question is a felony, the agent does not have to have witnessed its commission; they only need "reasonable grounds to believe" that the person they're arresting did it.

One possible felony that may come into play is 18 USC 1361: damage or attempted damage to federal property in excess of $1000 is punishable by up to ten years imprisonment. Anything greater than one year is a felony.