Rights don't apply when you are entering another country. Americans have the right to bear arms too, but good luck with that argument when get caught at the border with weapons.
I'm a bit skeptical that students are a big source of trouble. The vast majority come here, pay universities a lot of money, spend additional money in the local community, get their degrees, and then go home or maybe stay and work in generally high paying jobs, continuing to contribute to the local economy.
All that said, nobody has a "right" to come to the USA to study. It's something we allow, for a lot of good reasons, but there are doubtless a small number of people that we would not want here.
Do I think it's the best use of taxpayers' dollars (ie, mine) to screen for objectionable content on social media? No.
Do I trust the government to police opinions? No, especially when there's no accountability and appeals process.
Do I believe the overall benefits that harassment-free international travel brings to this country outweigh the costs of letting in some visitors whose views I disagree with? Yes.
Rights always apply, always. This is the thing about human rights enshrined in human rights laws in places like the EU, or about your constitutional rights (although the latter only applies to US citizens and to people physically present in the US).
However, countries may, depending on their laws, choose to not let certain people in on conditions that would otherwise violate guarantees on freedom of speech etc.
However, you do have your constitutional rights at the border etc. There is an exception concerning searches.
That depends a lot on the constitutional right. They're, generally, phrased as restrictions on the federal government (assumed to apply to state governments under incorporation post civil war).
There are a lot of times the government is limited even dealing with foreigners abroad (in legal theory anyways, ymmv in reality).
The courts haven't ruled on that yet but interstate commerce clause is pretty explicit about this kind of thing. A state can't criminalize someone going to another state to do something as that is something reserved to the federal government.
So no, there is no restriction on going to another state to get an abortion.
several states are trying to change that. laws are being proposed to ban travel, and although they are being challenged, as you said, the last word on this is not spoken yet.
and even if legal, people having an out-of-state abortion are being sued. that alone is a big restriction, because what good is a right if you don't have the means to defend yourself when that right is being challenged.
You are free to say whatever you like, as long as your words do not contradict Official Party Ideology.