Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham complex $$\Omega^\bullet_{X/Y} = \left[\mathcal{O}_X \xrightarrow{d} \Omega^1_{X/Y}\right]$$
This question discusses an exact sequence that relates the de Rham cohomology to sheaf cohomology, in the case when $Y$ is the spectrum of an algebraically closed field $k$.
I am looking for a reference for an analog of this sequence over an arbitrary base $Y$ (integral, Noetherian, whatever), which should look like this: $$0\to H^0(X/Y) \to R^0\varphi_*\mathcal{O}_X \to R^0\varphi_*\Omega^1_{X/Y} \to H^1(X/Y) \to R^1\varphi_*\mathcal{O}_X \to R^1\varphi_*\Omega^1_{X/Y} \to H^2(X/Y) \to 0$$
I would also like to be able to have such a sequence for more general de Rham complexes, like $$\mathcal{E} \to \Omega_{X/Y}(D) \otimes_{\mathcal{O}_X}\mathcal{E}$$ Where $\mathcal{E}$ is a vector bundle (or just line bundle), $\Omega_{X/Y}(D)$ is a sheaf of logarithmic differentials, and the map is an integrable connection. It would be great to weaken smoothness to normal crossing singularities. I will always consider the 2-term complex, truncated if necessary, so that the higher differentials $\Omega^i_{X/Y}(D)$ never enter the picture.
I figure that my desired sequence can be pulled out of some spectral sequence, maybe not even requiring anything to do with Kähler differentials. However none of the references I see spell out such an exact sequence, and I would very much appreciate seeing the details worked out.
