"Efforts to involve the U.S. armed forces in resolving election disputes would take us into dangerous, unlawful and unconstitutional territory," the former secretaries wrote in the letter, which was published as an op-ed by the Washington Post.
Edelman drafted and orchestrated the letter in consultation with former Vice President Dick Cheney, himself a former secretary of defense, along with help from former State Department adviser Eliot Cohen.
Cheney told Edelman he would sign the letter if he could get other former secretaries to join in, Edelman told ABC News on Monday. Edelman reached out to the Post and got all 10 secretaries to add their names by Friday, he said.
The motivation and timing for the letter was multifarious, Edelman told ABC News.
"There's the firing of Esper right after the election, there's the installation of this cadre of political appointees around (acting Secretary of Defense Chris) Miller there, there's the rush for the exit in Afghanistan," Edelman said, also citing a reported attempt by the Trump administration to split U.S. Cyber Command and the National Security Agency last month as well as Trump's controversial call with Georgia's secretary of state on Saturday.
Edelman also said comments made by former national security adviser Michael Flynn about the possibility of Trump invoking martial law to rerun the election in battleground states raised concerns.
In a Newsmax interview in mid-December, Flynn detailed what he considered Trump's military options. While Flynn claimed he was not advocating for these options to be exercised, and that constitutional processes must be followed, just weeks earlier he tweeted an organization's press release calling for "limited martial law" to hold a new election.
"I think Secretary Mattis for good reasons was a little reticent," the former official said. "He understandably feels that he as a retiree is still covered by the (Uniform Code of Military Justice), and you know, retired officers should not be criticizing the commander in chief, and they're not supposed to do that."
The former official told ABC News he came around after being persuaded by others involved in the project "that he needed to think about this not as 'former Marine four-star general officer Jim Mattis,' but as 'former Secretary of Defense Mattis.'"
Esper, who was ousted by Trump after the presidential election, was concerned about his participation looking like personal retribution, but quickly decided to join the other former Pentagon chiefs, the official said.
Chuck Hagel, a former Republican senator who served as secretary of defense under President Barack Obama, told ABC News Monday that he deliberated before adding his name to the op-ed to be sure it wouldn't be "making more out of something than maybe there really is."
Hagel said he decided Trump's actions and rhetoric aimed at overturning the results of the election posed a significant enough risk.
"I'm not overly concerned, but the fact is we have a president who has acted erratically, irresponsibly, and I think put our country in danger in many situations over the last four years," he told ABC News.
Former Defense Secretary William Perry said in a tweet Sunday that "Each of us swore an oath to support and defend the Constitution; that oath does not change according to party designation."
Robert Gates, secretary of defense under both Presidents George W. Bush and Obama, signed on without reservation when approached by Edelman, who served as under secretary of defense for policy during Gates' time at the Pentagon, according to a spokesman for the former secretary.
The letter concluded with a plea to the Defense Department to ensure a peaceful and smooth transfer of power.
"Acting defense secretary Christopher C. Miller and his subordinates -- political appointees, officers and civil servants -- are each bound by oath, law and precedent to facilitate the entry into office of the incoming administration, and to do so wholeheartedly," the letter said. "They must also refrain from any political actions that undermine the results of the election or hinder the success of the new team."
Edelman said this section was in response President-elect Joe Biden recently accusing political leaders at the Pentagon of "obstruction."
Miller pushed back on those claims in a statement last Monday.
"The Department of Defense has conducted 164 interviews with over 400 officials, and provided over 5,000 pages of documents -- far more than initially requested by Biden's transition team. DOD's efforts already surpass those of recent administrations with over three weeks to go and we continue to schedule additional meetings for the remainder of the transition and answer any and all requests for information in our purview," Miller said.
Some of the language in the letter directly mirrored comments from current defense officials, including Chairman of the Joint Chiefs of Staff Gen. Mark Milley and Army Secretary Ryan McCarthy, who have each said, before and after the election, that the military should play no part in handling election disputes.
"There is no role for the U.S. military in determining the outcome of an American election," McCarthy said in a Dec. 22 statement.
That line was quoted without attribution in Sunday's op-ed.
The other former secretaries who signed the letter are Ashton Carter, Leon Panetta, William Cohen and Donald Rumsfeld.