Over the years, I have found that some yoga practices have been helpful to people with asthma. Everybody is different and what works for one does not always work for another. From my own experience as well as reading, there is no one particular set of uniform practices.
No one should ever give up their medications and use yoga as a substitute for medical care! I suggest that these practices be tried when the breathing is fine, NOT during an asthma attack. If there is any discomfort, shortness of breath, tightness in chest, stop the practice immediately. The supervision of a yoga therapist is ideal.
Relaxing the respiratory system and its muscles is a good first step. Try a Yoga Nidra, which is done lying down. Use a comfortable pillow under your head and keep yourself comfortably warm. Keep both legs bent at the knees (keep knees hip-width apart) and the feet flat on the floor or bed. Any emotional triggers to asthma may work themselves out through a regular (3-5 times a week) practice of Yoga Nidra.
Learning to breathe right, through the nose, can also be helpful. Abdominal or belly breath, full yogic breath, and samavritti pranayama/balanced breath can facilitate nose breathing. Do not breath in ujjayi for any of them. These practices may be helpful in regulating the rhythm of the breath.
Some of my students have found kapalbhati pranayama (skull-shining breath) very helpful–but it is done slowly (15-20 per minute as opposed to 70-90 per minute) so there is no hyperventilation. Here the exhalation is active and short and the inhalation is passive.
Kapalbhati done in cobra (bhujangasana), or the sphinx, has been very effective for some. The chest opens up allowing for fuller breathing.
This can be followed by buzzing bee/bumble bee breath (bhramari pranayama) where the inhalation is shorter than the long, slow exhalation which is done with a slight drawing in of the belly. Again, the exhalation is active–but long instead of short as in kapalbhati. The link is to Calming the Storm track and it takes the listener through the pranayama after a short story.
There is an interesting article, Asthma Answers, in the Yoga Journal by Barbara Benagh that may be worth reading. The author had her own experiences and lists a set of breathing exercises that worked well for her.
The latest issue of the newsletter by The National Center for Complementary and Alternative Medicine (NCCAM) states it has found no evidence of significant improvement in asthma patients using breathing exercises or acupuncture.