Can army draft dodgers travel on a different passport?

There are two things which seem to be mixed up here.

“Dodging a draft” may describe a specific crime, where you explicitly promised/were ordered to serve in military, but you refused. This would be considered a crime, and it might be extradictable.

“Dodging a draft” may also refer to a generic practice of avoiding being in the position of being ordered to serve. For example, in Russia one cannot be prosecuted for dodging the draft until one has been personally served a prescription notice (“povestka”), and has signed for it. In this case “dodging the draft” is not a crime at all.

To add here, there are extra rules for expats. For example, Israel explicitly gives expats a break:

As a general rule, an Israeli citizen who has left Israel with both of
his/ her parents before the age of 14 (this age may be subject to
change by the Israeli authorities) or a child born abroad to an
Israeli parent (whose family has not returned to live in Israel) is
eligible for an army deferment (not exemption). An army deferment
means that the individual does not need to serve in the military as
long as he/ she is residing outside of Israel.

Thus just because someone dodging a draft doesn’t mean he’s committing a crime (and thus Interpol cannot be involved), and there is no need to even travel on a different passport (although you definitely can).

The Interpol angle is interesting. A red notice is supposed to be issued for “individuals sought for prosecution or to serve a sentence”. So if dodging the draft is a crime and the authorities care enough to prosecute it, it might lead to a red notice. It isn’t necessarily so, I know countries where proving you have fulfilled your military obligations prevents you from many things but isn’t actively prosecuted as a crime.

Furthermore, red notices are not supposed to be used for political purposes. Interpol has refused some red notices because the prosecution was deemed politically motivated but it has also been criticised for having weak standards and exercising insufficient control on how its member states use the system. But I don’t think this provision is very useful to draft dodgers in general, except maybe in some exceptional circumstances.

In practice, less than 10000 red notices are issued every year and about 40000 are in circulation. That’s not many for 190 countries and suggests that they are typically used for people suspected of serious crimes or sought for some other particular reason, not for the thousands of young men in your situation.

I know of two countries, with real-life cases: Korea and Singapore. Both in Singapore and Korea, a man can not renounce his citizenship until he has done his military service – even if he already has another passport.

In Korea it is even more perverse. A young Korean-X man, say Korean-American, can think he is American only, and still be considered a Korean citizen by the Korean government if someone in the family (usually it’s the paternal grandpa) registered his birth on the family register. That makes him automatically Korean. I know of cases where KorAm dudes came to visit Korea during summer and went home 2~3 years later, extremely fit and fluent in Korean…

There is a procedure, which changes, to defer military duty until you’re too old to do it. A friend of mine even managed to live in Korea, unmolested, back in the days when Korean-X kids were shipped to the Army for three years. But you have to be careful.

As for Singapore, the problem is doubly problematic: dual citizenship is illegal, and NS is compulsory. Which means you cannot go to Singapore, ever, if you’re an illegal draft-dodging dual citizen.

But that’s basically only applicable to a country – a Singaporean-American draft dodger can use his American passport to go to any country, except Singapore.

I believe that a generic answer is impossible.

• Conscription as such is not considered a human rights violation by Western nations, any many have extradition treaties with each other. In a slightly different example, the case of an US Army deserter in Germany is making his way through the legal system. So far he has lost in court, but IIRC there are still appeals pending.
• As a number of commenters pointed out, draft dodging is not as serious as the case of the deserter. Extradition should be somewhat less likely.
• Avoiding the draft of enemy countries is seen as praiseworthy. South Korea won’t return North Korean draft dodgers.