As an insight into the mind of left-wingers, a common answer was that the left doesn't support nationalism because it wants to help everyone:
Homiros: If you are a true left party, your main focus should be the welfare of all people.
FvHound: Us lefties don't want a compromise with the right. We want to help EVERYONE
Clearly, there are left-wingers who like to see themselves as compassionate types. Perhaps, therefore, an important aspect of challenging left-wing thought is to point out the hurt and damage that left-wing positions cause to many people.
Why would the left think of itself as helping everyone? Two possible causes spring to mind. The first is the political shift that happened in the late 1800s/early 1900s. The older liberals in the U.S. believed that the Anglo-Saxons had a special dispensation to bring freedom and Evangelical Christianity to the world. The new liberals challenged this view by rejecting the ethnic particularity and (often) by rejecting Christianity, in favour of a more ecumenical, cosmopolitan and humanistic view.
If you see yourself as serving "Humanity" in a cosmic sense, rather than God, then your focus will be a universal one (you won't be focused on helping your own nation, as the entity you have set up to serve is a global Humanity).
However, another possible explanation is that the leftist view is a cut-down, secularised version of a certain type of Christian ethics. If you think that the essence of morality is a vaguely universal command to help everyone including (or especially) the stranger or the marginalised, then you might well think that being a good person means, vaguely, a commitment "to help everyone".
It's true that the modern left has become cosmopolitan. However, what is less clear is that the move to vaguely universal commitments has oriented the left toward "serving everyone". The universalism seems instead to go along with a radical individualism, in which what matters is individual rights or the unfettered pursuit of individual wants or the liberty to define one's own good.
Furthermore, where the left is oriented toward community, it is often on an "assortative" basis, i.e. it is a gathering of people who share the same intellectual aspirations, the same political values and similar lifestyle markers. The left is very good at establishing community for itself on this basis - in some ways it has replaced the idea of ethny.
Why is this significant? Because when a communal identity is an ethnic one it means that we show a love for, and service toward, our coethnics who may not be part of our own caste or class: they may live in the countryside and not the inner city; they may eat fast food and drink beer; they may watch Channel 10 and not the ABC; read the Herald Sun and not The Age and so on.
But the left is not challenged in this way. In practice, leftists often show a disdain for those who do not share their own lifestyle markers - that is part of the way the boundaries of leftist community are upheld.
What traditionalists would argue is that we are not called to help everyone in a vaguely universal way. Our commitments are more bracing that that. We are made for particular relationships, relationships which imply specific loves and duties. Christianity does, it is true, remind us that our commitments don't stop at those we are most closely related to; this does not mean, though, that we are stripped of our given natures, or that we disregard natural law (or much of scripture) and seek to erase the significance of all those relationships bar the single, universal one.
For that reason a leftist should, just like any other person, seek to fulfil a relationship with a spouse, with children, with a wider family, with a local community (including those who do not aspire to intellectual class status) and with coethnics, as well as then helping others.