Cities are left wing. Rural is right wing. It would be a war on infrastructure, not a war on people. The right will, if it is smart, do its best to disrupt the logistics required to maintain the huge population density of cities. The left will play defense. If cities go into Venezuela mode because supply lines are cut, the leftist force will be split between peacekeeping on its own turf and trying to defend whatever critical infrastructure is left. Diversity in cities will cause factions to form along racial lines (or ideological lines if among ideological fanatics) as is the case in prisons and everywhere else you have a population isolated within adverse conditions. Tribalism takes hold. Every country is four meals away from revolution, and this logic would apply to each isolated city. Honestly, this whole situation is a nightmare for the left/urbanites. That would be the overall conflict.
"For the right, it would be absolutely critical to obtain Houston for fuel." For the left, it's not critical to hold Houston, denying it to the right is sufficient. What's to prevent some European collectivists from nuking Houston?
I am wondering how those leftists are going to GET to Houston. The initial stages of breakdown of civil order are likely to be, shall we say, chaotic and if communications degrade or break down I doubt there will be much opportunity a little further on down the road for identifying the mission, planning for, and training for it. The default is going to be existing local/rural advantages and that doesn't translate into leisurely advances to the objective. Local/rural advantages also includes cops, National Guardsmen, and a zillion vets.
But why Russia or China will support the RW if they hate them?