Not an answer, but too long for a comment: there are many details to adjust. I am not familiar with derived categories of rings, but I guess you are looking for an explicit description of the limit of $\infty$-categories along towers.

Firstly, recall that limits in an $\infty$-category arising from a model structure (as $\textrm{Cat}_{\infty}$ itself is) are computed as homotopy limits. Your diagram is not strict, but I think a rectification should always exist for diagrams of simplicial sets (as it does for topological spaces - see Vogt Theorem). Let's call $D(A_t)'$ the rectified version of your diagram.

Secondly, homotopy limits of towers can be computed nicely by taking a 'fibrant resolution'. Let us consider the Joyal model structure on simplicial sets. Since derived categories are $\infty$-categories, your functor is made of fibrant objects. If you look at proposition 3.4 in the nLab page above (or, equivalently, cor 2.4.6.5 in HTT), to prove that the diagram is fibrant we should also prove that $D(A_{t+1}) \to D(A_t)$ is an inner fibration and an isofibration (the same nlab page contains relevant definitions), and then transfer this information to $D(A_t)'$. An alternative approach to the homotopy limit computation, if the maps turn out not to be fibrations, is to use the so-called "Bousfield-Kan formula". It is an analog of the telescopic formula for homotopy colimits of cotowers of spaces: morally, it multiplies each term $D(A_t)'$ for a contractible simplicial set and then take an appropriate limit.

If we assume that the above is true, we get that the $D(A)$ is weakly equivalent to the limit of $D(A_t)'$. Since the Joyal model structure is Quillen equivalent to the Quillen model structure (via 'groupoidification'), weak equivalences should be also equivalences in the Joyal model structure. This means that when looking for a right adjoint departing from $\textrm{Cat}_{\infty}$ we can substitute $D(A)$ with $\lim_t D(A_t)'$, where the latter is meant to be the classical limit of simplicial sets.

At this point, the right-adjoint functor is probably given by the limit you thought, since an element of $D(A)$ is a sequence of compatible objects in $D(A_t)'$... I hope it works.