What is the relationship between international branches of the LDS Church? According to this story on NPR, Mitt Romney's campaign and his anti-Russian remarks have apparently been the impetus for Putin's "Young Guard" to dog Mormon churches in Russia, accusing them of being "a totalitarian cult" and that the LDS church in Russia is a CIA front organization. 

Obviously, I'm rather skeptical of the CIA front claim. I've worked intelligence long enough to be proud when someone accuses the CIA of being able to run anything.  Still, the claim is sufficiently reminiscent of the old claims that Catholics were supposedly beholden to the Pope in political affairs - enough to kill Alf Landon's chances, and sufficient to make JFK's election very, very tight.

The lie to the "Catholics are just political tools of the Pope" line, is of course, that there is an American Conference of Bishops that, while it reports to, is distinct from the Vatican.  Additionally, the American Catholic Church has very public disagreements with the Vatican on several issues.

My question is what analougues exist to the Russian (or any other countries') Mormon churches.  Politically, is there a national alignment of non-US Mormon churches, and what is their relationship to Salt Lake City?  Additionally, are there any large issues of disagreement that are more pronounced in countries outside of the US that distinguish it from the American leadership?

In other words, how do non-US Mormon churches relate to the US, generally?
They don't "relate to the US", except insofar as the headquarters of the church happen to be located in the US.

There aren't "international branches" of the LDS church; there is one worldwide church. The church is organized into individual congregations, known as wards or branches (a branch being a smaller congregation in an area where the church is not yet well established), and several local congregations located close to each other (generally around 10) make up a stake.  The presidency of each stake is chosen and ordained by a member of the Quorum of the Twelve Apostles.  Each stake is fairly autonomous within its boundaries, possessing the authority to direct the business of the church in its area, up to and including granting temple recommends (which are considered valid at any temple in the world) and holding disciplinary councils which can result in a member's excommunication, which again is binding throughout the entire church.

Several stakes make up an Area, which has an Area Presidency, though this is mostly an administrative grouping; church members have little if any interaction with the church at the Area level.  Areas are determined more by geography than national boundaries.  There are several in the US, for example, whereas when I was serving as a missionary in South America, the Area included all of Argentina, Uruguay and Paraguay.

At the highest level are the General Authorities of the church, who are headquartered in Salt Lake City.  Basic doctrines, directives and practices for the whole church come from there, and are accepted and followed in stakes, wards and branches throughout the world.

While stakes, areas and the Church Headquarters in SLC could be seen as roughly equivalent to the Catholic diocese, archdiocese and Vatican organizational model, there is no independent LDS organization equivalent to the American Conference of Bishops.  By the principle of Common Consent, the members of the church throughout the world--including local leaders--sustain and affirm their faith in the leadership of the General Authorities as God's chosen representatives on the earth in our time, twice each year at General Conference.  Under such a principle, there's really not much call for such an organization.

And one interesting thing in the linked article that deserves a bit of clarification is the following:


  The Young Guard contends that Mormon missionaries in Russia cooperate
  with the FBI and CIA. They repeat another theme that's common on the
  Internet as well: that American Mormon missionaries go to work for the
  U.S. security agencies in disproportionate numbers when they return
  home.


There's a lot of truth to that, but the reason behind it is much simpler than coming up with shadowy conspiracy theories about the church working for the US government (or vice versa, of course!)  The CIA and FBI (particularly the former) need people with a solid knowledge of foreign languages and cultures.  LDS missionaries in the US frequently get sent abroad and spend two years totally immersed in a foreign language and culture, a practice that was established long before the FBI or CIA even came into existence.  It just happens to be an excellent method of acquiring a core skillset for that particular type of job, (as well as several other jobs that deal in international business,) and agencies that look for people with that skillset when recruiting are aware of that.