Let $\mathcal{L}$ be the first-order language with equality that has a single binary predicate symbol $R$. Write down the following sentence, which I describe informally:
1) It is never the case that $R(x,x)$ and
2) $R(x,y)$ iff $R(y,x)$, and
3) for any $x$, there is a unique $y$ such that $R(x,y)$.
The sentence we have given a recipe for is satisfiable in any finite set that has a (non-zero) even number of elements, but not in any finite set that has an odd number of elements. We can modify the sentence by throwing in however many quantifiers that do nothing. Satisfiability of a sentence of this type in a domain of size $N$ does not imply satisfiability in a domain of size $N+1$.