AHh lots of fun to respond to....
First off the topic of wireless mics. The microphones used on the TED Talks is likely the DPA 4088.
Second on the topic of them being wired or wireless if they connect to a belt pack. You will not be able to get the power and quality required out of a headset unless you have something the size of a beltpack near it. Mind you there are some small beltpacks and it isn't unheard of for me to hide them in wigs on stage, see the Lectrosonics SM series for example. But they are all many times larger than the mics themselves to provide both quality, predictable and minimum latency, as well as required power.
I have put mics and beltpacks on dancers doing all sorts of manuvers. It takes some time to get right, but if you take your time, and think about things before hand you can secure them to instructors doing martial arts just as well. The catch is, depending on what exactly they are teaching, where you position the pack has to be carefully thought about, as one on the small of the back or waist won't work very well if they will be demonstrating basic breakfall techniques, etc. You would likely want to look at the inside of the thigh, placed there using a mic belt.
In as far as restriction of movement, this is one of many things that set a good A2 apart on broadway, they will take time to ensure that the cable is secured close to the body where it needs to be, with appropriate strain relief (On broadway this is likely tape, but you can use clips etc. as well in places where hiding it is not as essential) where needs to happen so that there is no restriction of movement.
In the end if an A2 has done their job right, the talent can forget they are even wearing a microphone. But also this requires a fair amount of prep work, and time with the talent. This may be something your instructors are unwilling to do. In which case your best option may be to use a shotgun or hyper cardioid, and follow the talent around. This may itself be a distraction however, so a choice is going to have to be made at where ever you are located and explained to the talent you have coming in to teach.