No, most countries will not allow their courts to assert original jurisdiction for torts or crimes committed in other countries by foreign actors. The US has more exceptions to this then most other countries, for instance the controversial https://en.wikipedia.org/wiki/Alien_Tort_Statute
But even the Alien Tort Claims Act only applies to torts, and only to those which are "...committed in violation of the law of nations or a treaty of the United States."
Other countries have a mixture of ideology and consequences which limit the scope of their jurisdiction.
The ideology itself formed mostly from historical consequences and collaboration with neighbors.
The consequences being their relationship with other countries. Specifically related to negative consequences that have an impact.
The US ideology has no limit to its scope, and the consequences are non-existent. When other countries disagree with US ideology, their desire to maintain a relationship with the US is greater. This allows the US to so far operate with impunity, with notable exceptions being Russia and China, which do ignore administrative requests from the US for things within their territory.
This should give a lot of color and context to why the US has a different meaning of jurisdiction than what people expect.
A future US could be subject to unilateral drone strikes and internationally coordinated sanctions, forcing more mutual cooperation and limitations in its enforcement scope.
I think it has less to do with drone strikes and more to do if if you want to do business in a given place and they have courts who take action, you're more attentive to the law there.
Nobody is worried about China drone strikes, but they'll hand over IP or whatever is needed.
But even the Alien Tort Claims Act only applies to torts, and only to those which are "...committed in violation of the law of nations or a treaty of the United States."