Should men who had relations with women for a long period and then got separated give financial support for the woman's children even if the children does not belong to the man?

No, the child's bio father should always be the one giving support, so these children have a bio father who is the responsible person.

Now from a moral standpoint, yes he should be there and help them if he can. But since he also has no legal right to even see them, often his ties are completely cut off once they separate

I agree with chuck. If he still has a friendly relationship with the ex and she lets him see the children, it would be nice to spend some money on the kids, or give her a few bucks every week to help out, but it's definitely not legally required.

should men who had relations with women for a long period of time and then got separated give financial support for the woman's children even if the children does not belong to the man?
If it is voluntary.

should men who had relations with women for a long period of time and then got separated give financial support for the woman's children even if the children does not belong to the man?
Blood is not thicker than water, does this man call these children his? Do they call him dad?
If he is a loving good father to these children he would want to make sure they are taken care of.

Broken relationships can have a dramatic effect on the children more than the anyone else. This type of situation is the very important reason, in my opinion, why it is vitally important for the biological parent to be held accountable for their portion of the child / children's financial well being.

Now, on the flip side of this: Yes, he can be held responsible:

There has been court rulings on just this situation under the Child Support Enforcement Amendments of 1984, P.L. 98-378, Section 18 as codified at 42 U.S.C. sec. 667. The state cognate is Chapter 460 of the Acts of 1993 (http://www.falseallegations.com/child-support-by-nonbiological-fathers.htm).

Although I am sure there must be a certain time frame set to argue this motion. JudyKayTee can probably lend a better insight to that aspect.

No, the father of the children is suppose to be doing that. And if she and he had followed God's plan they would be married and together and this whole problem or question would not exist.