"be being", especially when written "BE + being" represents: any one of the forms of the verb BE (i.e. am, are, is, was, were, (have) been followed by the word being. An example of this is: That child is being very difficult. The grammatical term for the form in this example would be the Present Progressive (or Present Continuous) Tense (or Aspect or Form) of the verb BE.

In over 45 years of being consciously aware of English forms, I have never encountered "do doing" But then I've never been to Boston