Such teambuilding sessions have two parts. The first part concerns training the team in the tools they will use: problem solving, statistical process control, flowcharting, etc. After an initial overview, this training is best delivered in a "just in time" fashion, where trainers teach the members the specific tool just before they use it. As an example, a team might receive an overview of problem solving as part of their initial teambuilding, but where they learn how to develop flowcharts just before they use them.
The second part of teambuilding involves some training in the usual set of group skills: meeting management, stages of group development, avoiding groupthink, the Abilene paradox, etc. For the most part, though, the second part involves facilitation around the specific issues (described above) that a particular team faces.
As you might guess, all this training/facilitation is best done when the entire cross- functional team is present in a room, all receiving the training/facilitation at the same time. Many companies do not realize this, and "mix and match" class room attendance, and train individuals from a variety of groups. This way may make the scheduling of training easier and more efficient, but it does not promote the spirit within a particular team. And isn't that the point of teambuilding?