As Democratic leaders tentatively took impeachment proceedings off the docket this week, the White House put payback on the front burner -- calling for closer looks into everyone from the FBI officials who investigated the Russia case to allies of Hillary Clinton's campaign who solicited foreign help during the 2016 presidential campaign.

“All those things have to be explored and more,” Trump lawyer Rudy Giuliani told "Fox & Friends" on Wednesday.

On social media and in televised interviews, President Trump, his attorneys, his campaign and senior members of his administration have in the wake of the Robert Mueller report seemingly adopted a strategy of highlighting lesser-known episodes of alleged misconduct by Democrats and investigators, as Democrats pursue obstruction of justice inquiries.

White House adviser Kellyanne Conway led the charge on Monday, openly wondering on Twitter why former FBI Director James Comey focused so heavily on the lurid and salacious claims in a largely discredited anti-Trump dossier.

"Comey, then-FBI Director, waited 2 months after @realDonaldTrump was elected to pay a visit & brief the President-elect," Conway wrote. "While there, he wasted his time on this golden-shower-nonsense-concocted-dossier. Could have been honest about Obama ignoring Russian interference instead."

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"Tables are finally turning on the Witch Hunt!" Trump also tweeted.

The FBI is currently being sued by conservative group Judicial Watch, after the bureau failed to respond to a Freedom of Information Act (FOIA) request concerning contacts in late 2016 between the then-FBI general counsel and a top Hillary Clinton campaign lawyer, as well as contacts between the FBI and the author of the dossier.

Comey briefed Trump on the salacious contents of the dossier in January 2017, ostensibly to make him aware of potential blackmail threats, Comey later testified. That confidential meeting later leaked, and CNN cited the fact that intelligence officials had briefed Trump on the dossier as a justification for airing the story, even though the dossier's claims were unverified.

Comey told lawmakers that then-National Intelligence Director James Clapper, an Obama appointee, came up with the idea to brief Trump on the dossier's contents.

Separately this week, Giuliani revived Republican-led calls to look into whether, and to what extent, a Democratic Party consultant worked with Ukrainian officials to dig up dirt on the Trump campaign.

"Keep your eye on Ukraine," he said Wednesday.

A 2017 investigation by Politico found that Ukrainian officials not only publicly sought to undermine Trump by questioning his fitness for office, but also worked behind the scenes to secure a Clinton victory.

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Among other initiatives, Politico found, the Ukrainian government worked with a DNC consultant to conduct opposition research against Trump, including going after former Trump campaign chairman Paul Manafort for Russian ties, helping lead to his resignation.

Last month, Ukraine Prosecutor General Yurii Lutsenko opened a probe into the so-called black ledger files that led to Manafort's departure, after a leaked tape recording apparently showed a senior Ukrainian anticorruption official admitting to disclosing Manafort's information to help the Clinton campaign. A Ukrainian court later ruled that the move amounted to illegal interference in the U.S. election.

"Now Ukraine is investigating Hillary campaign and DNC conspiracy with foreign operatives including Ukrainian and others to affect 2016 election," Giuliani tweeted Tuesday. "And there’s no Comey to fix the result."

On Sunday, Giuliani hammered the same theme. "Is it a crime for an American campaign to consider information from a foreign source or to obtain it?" he asked, responding to claims that the Trump team acted improperly by meeting with Russian-affiliated individuals who promised damaging information on Clinton.

"If so the allegation that the DNC [Democratic National Committee] colluded with Ukrainian officials to generate information to hurt the Trump campaign and help the Clinton campaign must be investigated," Giuliani added.

Buoyed by Special Counsel Mueller's findings that no member of the Trump team illegally conspired with Russia, Republicans have additionally turned to a separate known episode of apparent collaboration between a 2016 presidential campaign and a foreign national -- specifically, the decision by Hillary Clinton's campaign and the DNC to hire Fusion GPS. The firm, in turn, funded the infamous dossier, drafted by British ex-spy Christopher Steele, which contained numerous assertions that fueled an anti-Trump media frenzy -- but that Mueller's investigators were unable to substantiate.

Nonetheless, the FBI relied heavily on the dossier to obtain a Foreign Intelligence Surveillance Act (FISA) warrant to monitor former Trump aide Carter Page, and the bureau did not clearly disclose to the FISA court that Steele was working for a firm funded by Clinton and the DNC. Rather, the FBI told the court the materials were prepared in connection with a campaign for president. (Only partial versions of the FBI's FISA application has been released; Trump has told Fox News he will eventually declassify and release all relevant information from the FISA application.)

Democrats, too, heavily pushed the Steele dossier. At a 2017 hearing, now-House Intelligence Committee Chairman Adam Schiff, D-Calif., confidently described Steele as "a former British intelligence officer who is reportedly held in high regard by U.S. intelligence" and repeatedly cited "Steele's Russian sources" as he described a purported Trump-Russia conspiracy.

But in an article last week, The New York Times joined a chorus of publications that have long cast doubt on the dossier's veracity, writing that the document "financed by Hillary Clinton’s campaign and the Democratic National Committee" was "likely to face new, possibly harsh scrutiny from multiple inquiries."

The article noted that Steele relied in part on Russian sources and that, ironically, the document could have been part of a "Russian disinformation" effort to smear Trump even as Moscow was going after Clinton.

The article suggested that dossier skepticism, once panned as denialism, has entered the mainstream, as Mueller's report found "some of the most sensational claims in the dossier appeared to be false, and others were impossible to prove."

Internal FBI text messages obtained by Fox News last month showed that a senior Justice Department official warned of "bias" in a source key to a FISA application. The DOJ Inspector General is investigating whether the FBI violated its procedures or Page's constitutional rights by withholding exculpatory information from the FISA court.

"The Office of the Inspector General has a pending investigation of the FISA process in the Russian investigation, and I expect that that will be complete probably in May or June, I am told," Attorney General William Barr testified earlier this month.

House Intelligence Committee ranking member Devin Nunes, R-Calif., has already sent several criminal referrals to the Justice Department related to alleged crimes committed during the Russia probe, and Fox News is told as many as "two dozen" individuals could be implicated. It was unclear exactly whom had been referred.

"The American people have only seen the pieces that have been declassified so far," Nunes told Fox News earlier this month. "There's still more information."

The Trump team's pushback comes as top Democrats, including House Speaker Nancy Pelosi, have clamped down on calls to impeach the president. In a private Democratic conference call on Monday, two sources told Fox News, Pelosi said impeachment would be premature, and even anti-Trump firebrand Maxine Waters declined to call on her colleagues to begin impeachment proceedings.

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However, Fox News is told Democrats emphasized on the conference call that more investigations and fact-finding are necessary before any final decision can be made.

Schiff, for his part, has already referred to a June 2016 Trump Tower meeting as "direct evidence" of collusion in plain sight. Schiff has only doubled down since Mueller's report was published, calling for closer looks into Trump's finances and contacts with Russians.

Donald Trump Jr., his brother-in-law Jared Kushner, and Manafort were known to have attended the meeting with Kremlin-linked attorney Natalia Veselnitskaya, and the Trump team gave shifting explanations for the role of the president in drafting media responses to inquiries about the episode.

But Mueller found that the meeting was not a criminal campaign finance violation, in part because there was insufficient evidence that the involved parties knew they were breaking the law -- a high standard that applies only to certain crimes. Additionally, prosecutors said, it was unclear whether an exchange of information not available on the public marketplace could constitute a "campaign contribution" by a foreign national in the first place.

Still, Schiff told Fox News that episodes like that meeting raised grave concerns, and indicated impeachment would be a "difficult" issue that would be addressed in a matter of weeks.

In the dueling messaging wars, though, the Trump team has been nothing if not confident in recent days.

Giuliani told "Fox News Sunday," for example, that "so far we don’t think we need to" release a planned counter-report to Mueller's findings, because "we think the public debate is playing out about as well as it can -- why confuse it?"

Fox News' Adam Shaw and The Associated Press contributed to this report.