How are countries' militaries responding to inflammatory statements made by incoming US leadership?

Recently there have been multiple controversial statements made by the incoming US leadership and associates (i.e. Trump, Musk) towards other countries and their leadership:

Although this can be just words never to be backed up, just like campaign promises, the wildcard nature of Trump and co seems to show a willingness to act in an unconventional manner, making some of these words a (remote) possibility.

Additionally, the high profile nature of the individuals making these statements can cause great political effects in the target countries' population. They can and are increasing political polarity as has happened in the US, turning the political tide towards support for a relatively minor political party, and possibly destabilising support for a current government (e.g. Musk's attacks on UK's Labour government and Germany's Chancellor).

The question here is, although it is understandable that most is unlikely to be in view of the public, what signs are present and showing how the defence departments of countries are reacting to these statements?

For now the EU's civilian bosses have indicated they "would not put up with it":

Germany and France have warned Donald Trump against threatening Greenland, after the US president-elect refused to rule out using military force to seize Denmark's autonomous territory.

German Chancellor Olaf Scholz said "the principle of the inviolability of borders applies to every country... no matter whether it's a very small one or a very powerful one".

French Foreign Minister Jean-Noël Barrot said "there is obviously no question that the European Union would let other nations of the world attack its sovereign borders".

And Canada's soon-to-be-ex PM has said Canada would never join, which is in line with popular sentiment.

Trudeau, who has until now not directly commented on Trump's repeated and escalating comments about annexing Canada, appeared to have had enough.

"There isn’t a snowball’s chance in hell that Canada would become part of the United States," Trudeau said in a statement posted to social media.

The American electorate, in all its wisdom, has decided to elect a man who is somewhat of a loose cannon. Most governments will have to thread the needle on how to handle it. As long as Trump doesn't act on his "ideas", there is little to be gained by "provoking" him by pushing back too much. This is a man who repeatedly flares up at any personal slights he perceives and uses threats and intimidation as negotiating tactics. This is also a man who promised much and delivered rather less during Trump 1.0 (getting to replace SCOTUS positions is a boon that fell into his lap, bringing to mind Pearl Jam's "Born on third, thinks he got a triple") so the likelihood of those invasions is highly uncertain.

If I recall correctly, Japan's then PM was one of the better ones at managing Trump 1.0: public flattery, small, highly visible strategic concessions. Poland came up with naming some base Fort Trump too, IIRC. Canada should probably name a forest after Trump, we can always change that and rename a municipal dump after him once he's gone. Meloni is on good terms with him, that can help.

It sounds like I am joking, but I am not really. It is highly unusual to have such an unstable head of government and doubly worrying when they command the world's most powerful military and economy. Governments should not automatically assume that US policies will stem from realpolitik as opposed to the psychological traits of one person, or his pandering to his "base", and they should wait and see how things play out in practice, rather than inflaming the situation by overreacting. It's a good question, yes, but discretion might be the better part of valor for the near future.

To answer the specific request about military responses: in line with the rest of this answer, it is much, much, too soon for the militaries to respond, as opposed to the political leadership and the military is not in the business of taking the lead on diplomacy in the first place.

Mark Rutte, as head of NATO, would be expected to take on the military spokesman role. Looking for "Mark Rutte" Greenland Trump you'll find articles where the civs talk about Greenland and Rutte talks about NATO things like the 2% goal:

Euronews

Skynews

Four years later, this particular bottle of wine hasn't aged very well, but this is unlikely to amount to much. Remember that Trump ran partially on stopping the "forever wars" - a lot of the US public is very wary of foreign entanglements. Another thing to consider: the militaries in question are all part of NATO, the US is the top dog in NATO and Trump often vents on NATO and threatens to leave it. Getting him started on that path is a more likely outcome from overreacting at this stage than achieving much of import.

p.s. No, the Greenland defense boost isn't that Trump related.

Analysts say that the plan has been under discussion for a long time and should not be seen as a direct response to Trump's comments.

but Trump may be using his "trial balloons" to spur more spending

"I think Trump is smart… he gets Denmark to prioritise its Arctic military capabilities by raising this voice, without having to take over a very un-American welfare system," he added, referring to Greenland's heavy dependence on subsidies from Copenhagen.

(if not Trump, a much more likely cause for extra defense spending by Denmark, rather than what Trump said four years ago, or the day before, are the actions of a certain big Arctic state in the past few years... )