Perfectly corect if you want to go that far.
The DP is a temp to which a volume of air must be cooled at a constant temp.
As you say, the presure changes as the air is lifted, changing the DP temp slightly at altitude. That is as much as I can remember, but I know that for PPL-ATPL it isn't considdered.
I am making a guess but as Press reduces, so does temp (unless there is an inversion). If press reduces it would aid condesation and increase the temp the cloud would form, lowering the cloud base (beware of low press days) but it can only be a few feet....at a guess.