First this is clearly not the final answer and others may wish to post their thoughts.But your hazy question deserves an answer, with as much thought as a little time allows.
The certain connection between the white Kittrell and the black Kittrell is that they share the same surname.Nothing else can be said universally.By the time of the Civil War there were Kittrells in every southern state, but Virginia, some slave-owning and some not, and Kittrells further west and some Kittrells in the north from European immigration and some from southern immigration.
Any black Kittrells may have a specific basis for differing, but historically it is hard to get much proof of who was your ancestor absent the sanctity of heterosexual marriage (and that is rather rashly presumptuous---not the heterosexual part which was a given until this century, but that fidelity through marriage was assured).Each may respond to your query with a specific descendancy, but we all must rely on presumptions, some logical, some less so.
It is safe to presume that every black, using that term broadly to mean those of African descent and those of mixed African and other descent, with the name Kittrell here in America seems almost certainly to have obtained it from some ancestor who at one time was a slave of some white Kittrells (and in most cases are probably not related to that white Kittrell family or so I surmise).Some may be actually descended from a member of the slave-owner's family, but I have yet to hear the first story as to that.[Furthermore it must be emphasized there is not simply one tale as there were so many branches of this family, some in contact and most not, in so many diverse locales that each would have a separate version of what transpired up to the war if they were here to give.]
Before the war, there were a few mixed-race free folks (and a few non-mixed race black free folks) as well as mixed-race slaves (and many non-mixed race black slaves) of numbers and percentages never satifactorily delineated.In many small communities, at the turn of the century, the very parentage of a mixed race child of a not very mixed-race and evenly possibly single race black mother was known to the community and was usually the result of a non-marital relationship, among other reasons due to the miscegenation laws preventing anything else.J. Strom Thurmond comes to mind but Sen. Thurmond and parents may have help to avoid that problem perhaps, as the mother Carrie Butler, a Thurmond family househould maid, was most probably assisted by them and encouraged by them to give the baby for adoption to her aunt in far away Pennsylvania. [There is a strong resemblance between daughter and the late Senator.]
To summarize, each family, mixed as all families are, some by race, some by nationality and some merely by surname, must do its own sleuthing to find what applies to its own family's ancestry.
That is my version of reality, any one caring to provide theirs would provide you and me with welcome reading and perhaps provoke some new thoughts and theories.