Q
I'm wondering if the internal clocks in these devices are active only for the
conversion or if it is possible to disable them using one of the operating
modes?
A
There is no internal clock for AD7689 and AD7949. You have to provide an
external serial clock and datasheet recommends a discontinuous SCK because the
part is selected with CNV low, and SCK activity begins to write a new
configuration word and clock out data.
So you can disable external clock as per your requirements.