The above AKE principle implies the transfer principle between $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, and that one has some "real" applications.

The transfer principle states that for every first order statement $\phi$ in the language of valued fields, there exists a bound $N$ such that for every $p > N$ we have $\mathbb{Q}_p \models \phi \iff \mathbb{F}_p((t)) \models \phi$. To prove it, suppose the equivalence is violated for an inifinite set $P$ of primes. Let $K$ be a (non-principal) ultraproduct of the $\mathbb{Q}_p$ with $p \in P$ and let $L$ be the corresponding ultraproduct of the $\mathbb{F}_p((t))$. Then $K$ and $L$ have characteristic $0$ and they have the same value group and residue field, so AKE implies $K \models \phi \iff L \models \phi$, which yields a contradiction to the choice of $P$.

Here is one important application of the transfer principle:

The "Fundamental Lemma of the Langlands program" (which, actually, was a conjecture for a long time) is a statement about linear algebraic groups over non-archimedean local fields $K$. Ngo [arXiv:0801.0446] proved this "Lemma" in the case $K = \mathbb{F}_p((t))$ (and obtained the Fields Medal for that proof). Cluckers-Hales-Loeser [arXiv:0712.0708] showed that the transfer principle can be applied to that result, yielding the Fundamental Lemma for $K = \mathbb{Q}_p$ when $p$ is big. (Moreover, Hales had already shown before that it is sufficient to know the Fundamental Lemma for big $p$.)

Note that the Fundamental Lemma is an equality of integrals of functions from definable sets in $K$ to $\mathbb{C}$, so a priori, it doesn't seem to be a first order statement where the transfer principle could be applied. To apply it nevertheless, one has to encode the functions appearing in the Fundamental Lemma using model theoretic objects living purely in $K$ and then check that integration can be carried out purely on these encodings. In this way, the equality of integrals becomes a first order statement. (This can be seen as a very short explanation of what *motivic integration* is about.)