I am thinking of Eastern Europe, Central Asia and CIS-countries, Southeast Asia and other regions which recently underwent an economic boom
I have had some experiences in regions that would fall under these general groups. Although what I will say it is not at all representative of the entire situation it might give you some ideas. I think you are overall right. There are few things to be cautious about.
they may be more interested in hiring Western graduates from older/more reputable universities ... , and the faculty may consider employing researchers from more distinguished places a valuable marketing tool, at least in the beginning
This can be true. I have seen people get very excited because their department hired a top US school PhD without knowing much of what this person had done in their careers. I have also seen too many cases of western fetishism. Students/academics are eager to go to west for education/jobs. You can be a valuable bridge in that sense. There are huge differences between countries and their academic systems. For example university admissions are significantly different in different parts of the world. Oppurtunities like REUs are very common in US but they hardly exist, say, in Europe. Universities in these "second world" countries are often valued (by the public) on the rate which they send students abroad. Your background might be valued by default. You can provide your students very valuable insights into how US system works.
Industrialisation and population growth almost necessarily mean a higher enrolment rate and new institutions.
This is not necessarily the case. I know at least one example where the economic and population boom did not result in stronger institutions. What have happened is they have raised the quotas in Universities (almost all are public) without much increase in their budget. They have also formed new Universities but these almost always are pretty shady and corrupt. There are various department wide plagarism scandals. I have heard dreadful stories from my fellow student friends in these institutions. Furthermore, in the previously strong institutions, increased quotas and their substandard budget resulted in a visable decrease in education, research and campus life quality.
Just to be clear, I am specifically referring to second- or even third-tier target institutions, not Tokyo University or Peking Uni, though I welcome all suggestions.
I would suggest you to look at first tier universities at least in their respective countries. This will help you avoid few problems such as:
This need not be the case. Going to west is almost impossible (at least to a decent institution) unless you are increadibly wealthy in the second-world country standards. SATs, GREs, application fees are huge when the minimum wage is around 300 dollars. Not to mention that these countries will NOT have school systems built around these exams. These people will not have any experience in getting tested in English or understand how the admission game works. The US oppurtunity exists for the very few. These are usually the extremely wealthy who can afford private schools and private tutors aimed in this direction and the extremely smart if they have a conventional way to prove it (like mathematical olimpiads). Even at graduate level capable people hessitate to leave their country or return back without completing their phd abroad. There can be many cultural and economic reasons for that. Humans tend to be philopatrical. There is a huge between going to California from Alabama and going to US from Malaysia. Especially after the current wave of Islamophobia. But yes, overall expect weaker students. But I would bet the good ones you are going to get will be pretty good.
This is true on avarage but an elite institution of a second world country can easily have more resources than a second rate US university. It is really case specific.
do you have any special advise on the application process or on the difficulties I may encounter
Difficulties vary but you should expect some. For example the country may have strict regulations for Universities. The titles might be subject to certain criterias (x many publications, y many books etc.). These are unpleasant. I have seen top tier researchers getting stuck with a lower title not because the university does not recognize them. Because the country will have some weird numerical criteria attached to that title.
Is the administration more likely to welcome me among their ranks at such places, ceteris paribus
This again is very case specific. If you get an offer from a university I would suggest you to get in touch with people in that university first. My institution is pretty decent in that regard. I would say at least half of the faculty is foreign and around two thirds of it are post-Soviet / East Bloc academics. Although I wouldn't be suprised if some other institution's department head / dean / President would have an issue with a certain foreign nationality.
One key point that you have not mentioned is political unrest. In Hungary, Orban forced out Central European University. In Turkey, Erdogan did serious academic purges. In US, misinformation and pseudo-science are spread by politicans including the Mr. President. Countries like Iran, Indonesia, Libya, Egpty are still heavily shaken by their previous political struggles and many of their strong academics have fled the country.
I think you should consider such oppurtunities but you should be cautious. Research the institution well. Is there plagirism, fraud, nepotism, sexual abuse, bullying or censor? Is the institution reputable in its own country? Is the research aspect valuable or is it just a glorified high school with more advanced topics? You should try to understand the countries policies towards the academica as well as public opinion towards higher education. For example, Indian public seem to value humanities far, far less than engineering. In that case you might have a harder time in India as a humanities researcher. Or if politicians look at school budgets and drool then those institutions might be in slight trouble. Most notably in US some state legislatures cut their State's funding in very small portions year by year. Now, state universities are also a part of the University cost crisis. US economy is huge and they so far have survived this crisis with somewhat managable consequences. I am not so sure how it would play out in a much poorer country.