In this paper, we show some Second main theorems for linearly nondegenerate meromorphic mappings over the field $\mathcal P^1_c$ of $c$-periodic meromorphic functions having their hyper-orders strictly less than one in $\mathbb C^m$ intersecting slowly moving targets in $\mathbb P^n(\mathbb C)$. As an application, we give some Picard type theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ under the growth condition hyper-order less than one.