Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$.
There are several presentations of the Dieudonné module of $\mathbb G$: $\newcommand{\G}{\mathbb G}\newcommand{\A}{\widehat{\mathbb{A}}}\newcommand{\co}{\colon\thinspace}\newcommand{\W}{\mathbb W}\DeclareMathOperator{\Lie}{Lie}$
Cartier's presentation is as the module of $p$–typical curves on $\G$, i.e., maps of formal schemes $\gamma\co \A^1 \to \G$ which are annihilated by the prime-to-$p$ Frobenius operators.
One of Grothendieck's presentations is as the submodule of cohomologically-translation-invariant classes in $H^1_{dR}(\widetilde{\G} / \W_p(k))$, where $\widetilde{\G}$ is any lift of $\G$ to the ring $\W_p(k)$ of $p$–typical Witt vectors over $k$.
Another of Grothendieck's presentations is as the group of "rigidified" extensions $E$ of $\G$ by $\G_a$, where "rigidification" is the extra data of a splitting of the Lie algebra sequence $$0 \to \Lie \G_a \to \Lie E \to \Lie \G \to 0.$$
These presentations do not obviously all yield the same result. There are very explicit constructions in the literature comparing the second and third, where, e.g., a representing cocycle of a translation-invariant class is used to construct a specific extension. The first presentation is more dramatically different: it is naturally covariant, whereas the second and third are contravariant.
One bridges the differences between these presentations as follows. Curves on $\G$ can be identified with formal group maps $\widehat\W \to \G$, where $\widehat\W$ is the "formal Witt scheme" — i.e., $\widehat\W$ has the freeness property $$\mathsf{FormalGroups}(\widehat\W, \G) \cong \mathsf{FormalSchemes}(\A^1, \G).$$ There is even a formal subscheme $\widehat\W_p$ selecting just the $p$–typical curves. This lets us fabricate a replacement for "$p$–typical curves" on much smaller group-schemes (like finite ones) by instead taking maps into them from $\widehat\W$. Then, the contravariant Cartier-presented Dieudonné module of $\G$ is defined to be group-scheme maps from $\widehat\W_p$ to the Cartier dual $D\G$ of $\G$, now considered as a $p$–divisible group. Equivalently, one can put the Cartier dual "on the other side" and study homomorphisms from $\G$ to the "co-Witt" group-scheme $\widehat{C\W_p}$.
It's now possible to try to find a natural isomorphism between this modification of the first presentation and the others. Katz writes down such a map as Equation 5.5.2 in Crystalline cohomology, Dieudonné modules, and Jacobi sums by $$\begin{align*} \mathsf{FormalSchemes}(\G, \widehat{C\W_p}) & \to H^1_{dR}(\widetilde\G / \W(k)), \\ (..., a_{-n}, ..., a_{-1}, a_0) & \mapsto d\left(\sum_{n \ge 0} (\widetilde a_{-n})^{p^n} p^{-n} \right), \end{align*}$$ where the sequence on the left is written in "ghost components" and then $\widetilde a_{-n} \in \mathcal{O}_{\widetilde{\G}/\W_p(k)}$ is any lift of $a_{-n} \in \mathcal{O}_{\G/k}$. Lastly, he specializes to the appropriate subsets of the left- and right-hand sides to get the desired natural isomorphism of Dieudonné modules.
This last formula is very mysterious to me. Where did it come from? For that matter, where does the proof that this gives an isomorphism appear in the literature? Most importantly of all...
Question: What is a geometric description of this duality pairing?
In my wildest fantasy, this duality pairing would be motivated by something as simple as "Well, we would like to imagine integrating this de Rham differential along this curve...", and then the story would balloon on its own into this Witt scheme stuff. Any kind of slogan like that would be really excellent to hear.
