Threads, tasks, and the Objective-C SDK

I have been studying these API calls, and even with a custom queue, I don't see how to do what I want.  First of all, here is the pseudo-code for what I have implemented in Android (omitting all error-handling), and what I want to do in iOS.  It downloads all the files in Dropbox and runs in a single background thread - not the UI thread.  I have also omitted all interactions with the main UI thread:

Code for thread:
  processWithRecursion(");        //..start with the root of Dropbox
end of execution of thread.

And here is the recursively-called function, processWithRecursion():

  processWithRecursion( currentDirectory )
  {
    form list of currentDirectory
    for each element in that list, do:
    {
      if(element is a directory)
      {
        create local destination directory of that name
        processWithRecursion(currentDirectory + subdirectory name);
      }
      else  //..element is a file
      {
        create destination file and download from Dropbox to that file
      }
    }
  }

As you can see, the recursive call nicely steps through all the files, including any nested subdirectories.  The call stack performs the function of keeping track of where we are in the search for files.  This could be performed by an elaborate state machine, but then I would have to create a separate push-down stack data structure to traverse the directory tree.

This process is best implemented with API functions that block.  But it seems the API functions available to me do not block, so there is no easy way to execute a series of these API calls by simply writing them one after another in code.  For example, I would need to call listFolder:searchPath to get the list of files in one directory, and then for each of the items returned in the list I would need to call downloadData:filePath to actually do the download.  But both of these API calls are implemented as non-blocking calls that return immediately, leaving the completion block to handle the completion of the action.  So I could not write listFolder:searchPath followed immediately by downloadData:filePath, because the downloadData: method would start executing before the searchPath: method was completed.  In fact, I couldn't even form a filePath parameter for downloadData: without examining the completion of searchPath:

That was just for downloading one file.  When you throw in the need for traversing the entire directory tree with recursion, all without any UI input, I don't see a neat way of doing this.