The decision of whether or not to employ circuits in training depends on several factors, including:
What is the goal of the training session? Movement mastery, cardiovascular training, muscular endurance, speed and power, etc.?
How specific is this training meant to be? Specificity here refers to how well the training matches the actual sport (in this case, calisthenics).
What stage of the over-arching training cycle or career are we in? Off-season, competition/exhibition preparation, no particular season?
The Movement Athlete is a training platform which is primarily concerned with the learning, training and mastery of calisthenics movements. Many of the movements themselves are adaptations of gymnastics movements. The most effective and sensible way to learn and get used to the motor patterns, muscle activation and techniques required for these movements is to give each movement enough time, space and attention. In this way, the athlete can compare repetitions of a movement, tweak technique and immerse themselves in that movement without the distraction of other movements. In some cases, doing two movements back to back can even be dangerous, as the technique cues from one movement can interfere with the other.
In terms of muscle building and endurance training, circuits (here defined as a series of more than two exercises; supersets or antagonist sets which involve two exercises done back to back are not considered as a circuit) are a possibility. However, even here, separating the movements has its benefits. For muscle building, it can be safer when the joints and muscles for a particular movement are warmed up, kept warm and used while warm before moving on to another exercise. As with training movements, exercise quality and technique are also important in building muscle. Performing all of the sets of a particular movement together allows the athlete to better engage the muscles involved.
Circuit training has its place and can be a useful tool:
In sport-specific training, for example specific circuits for basketball players which mimic a part of the game.
To save time, commonly used in martial arts schools or gymnastics classes, where a bunch of conditioning exercises are done in rapid succession for 10-15 minutes after the class.
When low impact, low complexity movements are involved, or when the athlete is advanced or proficient enough that the movements are effortless and second nature.
To sum up, although circuit training may have its uses, for the vast majority of cases in The Movement Athlete training system, more appropriate methods have been selected.