The descent of the M2/M5-brane charges to the Dp-brane charges and their twisted Bianchi identites dFp+2=H3∧Fp is straightforward for p∈{0,2,4}, as spelled out rigorously for instance in section 4.2 of
Varghese Mathai, Hisham Sati, Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP0403:016,2004 (hep-th/0312033)
The origin of the D6 brane charge F8is more subtle, as its M-theory lift involves the field of gravity: F8=⋆10F2. The correct Bianchi identity dF8=H3∧F6 is claimed on the left of (4.13) in the above article, but it does not actually follow from the 11d derivation given there, as far as I see.
What is a quick way to derive the Bianchi identity dF8=H3∧F6 in type IIA from 11d?