So what is the difference between being an asylum seeker and being a deserter?
A straightforward answer is that a deserter* can claim asylum, but being a deserter is not enough to claim it successfully. Not every deserter is eligible for asylum and not every asylee is a deserter†.
But, as far as I can tell, you're most concerned about the refugees escaping war-torn regions, where the lines become blurred. To address this properly I'll consider two different scenarios:
- The US is relatively stable and is fighting a war overseas. In this case, deserters are on the run from American law enforcement.
- The war is happening on American soil and the US government is on the brink of collapse. In this case, deserters are running away both from their own government and from the horrors of war.
Escaping a stable country
The actual requirements depend on the country, but usually, to claim asylum, asylum seekers need to present evidence that they face persecution for social or political reasons in their country of origin. Just being on the run from law enforcement is usually not enough. Some countries might recognize mandatory conscription as persecution, but most won't.
In the US, for example, asylum seekers need to fill form I-589 Application for Asylum and for Withholding of Removal. The definition of asylum is given in the instructions:
To qualify for asylum, you must establish that you are a refugee who is unable or unwilling to return to his or her country of nationality, or last habitual residence if you have no nationality, because of persecution or a well-founded fear of persecution on account of race, religion, nationality, membership in a particular social group, or political opinion.
This means that you must establish that race, religion, nationality, membership in a particular social group, or political opinion was or will be at least one central reason for your persecution or why you fear persecution.
As you can see, "dodging the draft" and "being on the run from law enforcement" are not listed as valid reasons to claim asylum. If the host country has laws similar to the US, American deserters in your scenario are unlikely to be granted asylum unless the host country decides to accept them anyway for political reasons (e.g. North Korea).
Moreover, the host country is likely to run a background check on asylum seekers in cooperation with the US government. Being wanted in the US for crimes, including desertion, can be a good enough reason to deny asylum and to deport an individual back to the US.
Another issue is that the US government will likely try to stop deserters from escaping the US territory. This will force them to cross the border illegally. If the host country has laws similar to the US they can be prosecuted for illegal crossing and subsequently denied an asylum.
Escaping a war-torn region
If the US is losing the war and/or otherwise becoming unstable, deserters might claim that they are no longer able to protect the US and at the same time fear the persecution from invading enemy/winning side of civil war. This way they will fit the definition of a refugee according to 1967 Protocol:
A person who owing to a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion, is outside the country of his nationality and is unable or, owing to such fear, is unwilling to avail himself of the protection of that country; or who, not having a nationality and being outside the country of his former habitual residence as a result of such events, is unable or, owing to such fear, is unwilling to return to it.
The success of their claim largely depends on the US being recognized at an international level. If at this point the host country still recognizes the US government it might choose to deny their refugee status and return them back to fight for their country.
Some other things to consider:
- Armed conflict of such a scale is likely to displace the large populations across the US. It might be a challenge to distinguish between deserters and displaced civilians.
- Unstable government is less likely to produce clear conscription records.
- In the case of civil war, the winning side might claim that everyone on the losing side is a deserter.
So, yes, people can apply for asylum if they think that the US or their interpretation of the US no longer exists. Something like this has already happened before, see Confederados.
To answer your other questions.
Is it generally accepted that citizens have the right to escape any war by running away to a foreign country?
No, citizens are expected to protect their country. It's generally accepted that for them to do so their country has to exist and recognize them as citizens. It's also generally accepted that they can claim refugee status if their government, winning side of a civil war or occupying force targets them for political reasons.
Do any governments somehow acknowledge this conundrum in their conscription laws?
As far as I can see, there's no conundrum. Other countries are unlikely to grant refugee status frivolously.
In general if your country:
- has an unstable government that is no longer recognized;
- politically persecutes its own citizens, including army recruits;
- forces the significant part of its military personnel to seek asylum in other countries
then you have bigger problems that can't be solved with conscription laws
* — As others already noted draft evasion is a separate crime, for the purposes of this answer I'll assume that we're talking about actual deserters. It doesn't change much.
† — This statement is so general it will work for any other pair of largely orthogonal concepts. You can replace "asylee" with "vegan" and "deserter" with "jaywalker" and it will still work.