I'm going to assume we're talking about $\omega$-incomplete ultrafilters.

Others have pointed out that none of the major dividing lines are elementary; that is, these properties are not equivalent to a countable conjunction of sentences.

Some dividing lines, like stability and NIP, have the form "for each formula $\phi$, there is a numeric n such that a sentence $\sigma_{n,\phi}$ holds". For instance, NIP holds if each $\phi$ has bounded VC dimension. Such a property holds in an ultraproduct exactly when it holds *uniformly* in the ground models.

That is, $\prod_{\mathcal{U}}M_i$ is NIP if and only if, for each formula $\phi(x;y)$, there is a bound $d$ such that, for most $i$, $\phi$ has VC dimension $d$ in $M_i$. That means an ultraproduct of NIP models can be IP (if the VC dimension is unbounded) and an ultraproduct of IP models can be NIP (if each model has an IP formula, but each individual formula is IP in most models)

I'm not sure if every dividing line can be expressed in that form. But (at the risk of self-promotion) there's a more general framework for describing properties of ultraproducts in terms of "higher order uniformity", so it will, in general, be true that a dividing line holds in the ultraproduct if it holds uniformly in the ground models, for a suitable notion of uniformity.