If $(X_1,X_2,X_3)$ is a gaussian vector, is $(X_1-X_2,X_3-X_2)$ a gaussian vector ?
I would say yes but I'm not sure. For me, $(Y_1,...,Y_n)$ is a gaussian vector if all linear combinaison is a Normal r.v. So if $(X_1,X_2,X_3)$ is a gaussian vector, all linear combinaison is a normal r.v. Therefore, $$\alpha (X_1-X_2)+\beta (X_3-X_2)=\alpha X_1-(\alpha +\beta )X_2+\beta X_3$$ is a normal r.v. and thus all linear combinaison of $X_1-X_2$ and $X_3-X_1$ is a normal r.v. and thus $(X_1-X_2,X_3-X_2)$ is a Gaussian vector.
Is this correct or not at all ?
Yes, your reasoning is correct. A clean proof which makes use of the same fact about preservation of Gaussianity under linear transformations is to notice that $$\begin{pmatrix} X_1 - X_2 \\ X_3 - X_2 \end{pmatrix} = \underbrace{\begin{pmatrix} 1 & -1 & 0 \\ 0 & -1 & 1 \end{pmatrix}}_{= A} \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix}, $$ so if $\begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} \sim \mathcal{N}(\mu, \Sigma), $ the resulting vector is also Gaussian distributed as $\mathcal{N}(A \mu, A \Sigma A^\top)$.