Adho Mukha Svanasana
Downward Facing Dog is one of the first poses the new yoga student in 21st century United States learns. However, the method used to execute the pose, and the importance of the pose within the tradition varies significantly across styles. In fact, early yoga texts didn't mention the pose at all, and some practitioners from as recently as the seventies are often surprised to find it become so important in practice today.
Here are four very different yoga styles and how they each practice adho mukha svanasana (downward facing dog). These variations in hold, breathing technique, use of repetition, or underlying sequencing theory, give examples of what factors characterize each style.
Iyengar
B.K.S. Iyengar performs Adho Mukha Svanasana, ca. 1991, from https://yogastlouisblog.wordpress.com/)
The paragraph accompanying this photo states that B.K.S. Iyengar practiced this pose for 15 minutes or more. Note that the aim here (not often taught now due to potential neck injury) is to touch the head to the floor. Notice also the distance between hands and feet, which is significantly longer than in other traditions.
Iyengar Yoga is known for long holds, in order to engage the subtle body, and for adjustments and modifications using props, as seen below.
(from http://www.eldr.com/article/fitness/healthy-a)ging-master-pose-downward-facing-dog
Ashtanga
(From http://www.ashtangayoga.info/practice/surya-namaskara-a-sun-salutation/item/adho-mukha-shvanasana/)
The attention in this tradition is less on the detail work in the pose, which is itself a small part of a much larger sequence set to a particular breath pattern and style of breathing (called vinyasa), although other breath techniques may be employed at other parts of the practice. Movements between poses are given greater significance than in Iyengar style.
(from http://www.mysoreyogacph.dk/news/yoga-lab-jump-back-jump-through/)
Viniyoga
from http://www.wholelifeyoga.com/blog/extending-the-spine-in-downward-dog/
Kundalini
(From http://www.yogajournal.com/slideshow/recover-holiday-feast-8-detox-poses/#slide-4)"Donkey kicks"
Like Viniyoga, Kundalini sometimes uses a pair of movements and alternates between inhalation and exhalation, often for a specified number of repetitions, with periods of rest in between. However in Kundalini, the main breath is much more rapid than in Viniyoga, so the movements are more active. Additionally, asana (poses) are usually specifically combined with other elements of yoga, such as mudra, mantra, and a point of focus for gaze.
* Note: The image above, Donkey Kicks are alternated with Adho Mukhasvanasana, and are not a transition. In Ashtanga, a similar technique is used to practice coming to an inverted half-handstand position in preparation for a "jump-through", swinging through the hands into dandasana, staff pose. A version of this completed technique is shown here, although from right to left.
(from http://www.yogayork.com/Ashtanga%20Improvers%20and%20Beginners.htm)
Summary
Iyengar: Precision in alignment, long holds increasing over time, use of props to modify. Sequencing is specific, but poses are still viewed independently rather than as just a component in a longer sequence.Ashtanga: Precision in transitioning movements are a priority, holds are generally a breath, making props hard to use in such little time. Sequences are firmly set and have been practiced in the same order for decades.
Viniyoga: Associated with Yoga Therapy, this style is focused on relaxing the central nervous system. As such, it tends to involved slow, deliberate movement, alternating between 2 poses for each inhale/exhale pair. Sequences are often a pair repeated 8 times before resting with a normal breath.
Kundalini: A style that employs use of additional techniques such as chanting, hand shapes, and rapid abdominal breath, poses are downplayed and taught in pairs with specific additional techniques. This complete sequence of alternating movement is called a kriya. Kriyas are often very active and focus on endurance, although classes are spiritual and playful.
This color wheel by the former Alison Hinks (now married & her site has moved) shows the relationships between different styles of yoga.
(from http://www.mindbodygreen.com/0-2739/Yoga-Wheel-Infographic.html)