An extension of Szemerédi's Theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg's Correspondence principle, which should be of independent interest, we show that our Szemerédi Theorems can be deduced from a general \emph{transverse} multiple recurrence theorem, which we establish using recent works of Austin.