Even with an adaptable chuck, the Dremel can only grip a certain size shaft, such that not all bits that can be bought are also compatible. Fortunately, there is an easy way to make the bits compatible using heat-shrink tubes.
The process involves adding layers of heat-shrink tubing on top of each other, every time using a heat gun or heat source to make them shrink, up to the point that the bit fits perfectly into the chuck.