Mr Flynn has long been a part of the Trump campaign. A former head of US military intelligence, he has direct experience of fighting the 9/11 Wars in Iraq and elsewhere. He clashed with the Obama administration over its insistence that Islamic militancy was on the wane and was forced out in 2014, becoming a noisy critic.
Mr Flynn was partially right about the resurgence of Islamic militancy, but his language since has been bitter and intemperate, with attacks on the "politically correct" failure to explicitly name the Islamic faith as source of the problem of extremism. He believes the US is engaged in a multi-generational "world war" against Islamic militancy which it is losing and, like Mr Trump, admires Mr Putin. He has called for regime change in states such as Pakistan and Iran.
Mr Pompeo lacks Mr Flynn's hands-on experience, though has served on Congressional committees dealing with intelligence. He has called Muslim imams, who he says have been insufficiently vocal in condemning violence, "complicit" in terrorist acts and is a staunch supporter of torture. He does not want terrorists treated as "ordinary criminals". Like Mr Trump, he is resistant to sending large numbers of US troops overseas to counter the threat and, unlike Mr Flynn, views Russia as a significant danger.
Though other officials have significant roles, these two men will strongly influence how the president runs the campaign against Daesh and Islamic militancy. The result will probably look very like the "Global War on Terrorism" (GWOT) of George W Bush's first term - excluding the invasion of Iraq.
The new GWOT is likely to see a mixture of vigorous if unpredictable action with limited regard for US or international law - so more drone strikes, bombing campaigns, special forces raids, assistance to local military actors deemed friendly and collaboration with agencies with scant respect for their human rights records. The CIA is likely to be encouraged - empowered even - to use torture.
But little of this is likely to be systematic. Torture will be used where agents on the ground share the views of their political and institutional superiors. Bigger interventions, particularly those involving large numbers "boots on the ground", will not be ruled out but will be haphazard and reactive, as befits an administration with strong isolationist tendencies.
A trial ground of the new GWOT is likely to be Libya, where Daesh forces are still holding out in Sirte. Mr Pompeo established his hawkish credentials investigating, and talking about, the killing of a US ambassador in Benghazi in 2012 and the alleged culpability of Hillary Clinton. There will be an early temptation to show in Libya how Mr Trump will succeed where Democrats failed.
The strategic map of the Levant is likely to be dramatically reshaped within six months when Mosul and Aleppo return to Iraqi and Syrian government control respectively. A new phase in the Syria- Iraq conflict will open. This, along with the apparent proximity of Mr Trump, and Mr Flynn, to Moscow, will make a future settlement ensuring the survival of the Syrian regime, backed by Iran and helped by Hizbollah, more likely. The meagre assistance to the "moderate" Sunni opposition factions will be cut off.
Any deal leaving Assad in place, of course, would cause problems with Sunni states across the region. Relations between Ankara and Washington have soured over recent years. Mr Trump, Mr Flynn and Mr Pompeo will be less bothered by Tayyip Recep Erdogan's authoritarian tendencies than Mr Obama's team, but much more bothered by his Islamism.
Other rulers will get a warmer reception. Both Mr Flynn and Mr Pompeo are likely to welcome Egypt's President Abdel Fattah Sisi, who Mr Trump has already called a "fantastic guy". Mr Sisi, like other autocrats, presents himself as a bulwark against Islamic militancy. At the White House he will be pushing at an open door.
The Gulf is likely to be trickier. Mr Trump's Islamophobic rhetoric has alienated many rulers already. It is hard to imagine Mr Flynn receiving a particularly warm welcome in Riyadh. But then the new NSA, and his boss, are both fierce opponents of the nuclear agreement signed by the US and five other states with Iran, and relations between the CIA and their Saudi counterparts have long been warm. The Saudi Minister of the Interior, Crown Prince Mohammed bin Nayef, has been an important ally of the agency for years. Both parties here have much to lose if they cannot hold their noses and get on with killing or capturing extremists.
And Israel? All of Mr Trump's advisers are deeply pro-Israel, and both Mr Flynn and Mr Pompeo are likely to support the most hawkish of Israeli policymakers without any reservations.
But the broader question is the relaunched GWOT makes Israel, the UK, Europe or the US safer. A century or more of such violence has taught us that waves of terrorism follow a relatively predictable pattern of rise and fall. The cycle - a major surge of violence, a steady intensification, a plateau then turning point and decline - usually last 15 to 20 years.
What allows us to reach the turning point is much debated. One element is certainly killing key leaders. Al Qaeda was "hollowed out", in the words of one MI6 official, even before Osama bin Laden died in 2011. But many other factors come into play. There is the denial of territory. As multiple jihadi strategists have theorised, Islamic militants cannot thrive without some kind of base. Lone wolf attacks are insufficient to achieve wider aims. Deny the base and control of population, and capacity to harm locally or globally is much reduced, as we saw with Al Qaeda in Afghanistan and are likely to see with Daesh in Iraq in the coming months.
Another key factor is community support, or at least acquiescence, both at a local level and a global one. This goes for all terrorism - whether in Ireland, the Basque country, Gaza or Syria. The turning point in last decade coincided with the moment when public opinion in much of the Islamic world turned against al'Qaeda.
The left tend to stress "softer" factors; the right "harder" ones. The truth is both hard and soft working together get best results. Alone, neither can succeed, and risks making the problem worse. Mr Flynn and Mr Pompeo, and probably the president-elect, would no doubt disagree. We can expect heightened rhetoric which will play into the hands of the militant propagandists, and uncompromising hardline tactics, which will anger and thus radicalise many. The result is unlikely to make us all any safer. As was learned in Afghanistan and Iraq, there is no purely military solution to Islamic militancy, however much Mr Trump's new team would like there to be one.
