In this paper, we analyze the finite-length performance of codes on graphs constructed by connecting spatially coupled low-density parity-check (SC-LDPC) code chains. Successive (peeling) decoding is considered for the binary erasure channel (BEC). The evolution of the undecoded portion of the bipartite graph remaining after each iteration is analyzed as a dynamical system. When connecting short SC-LDPC chains, we show that, in addition to superior iterative decoding thresholds, connected chain ensembles have better finite-length performance than single chain ensembles of the same rate and length. In addition, we present a novel encoding/transmission scheme to improve the performance of a system using long SC-LDPC chains, where, instead of transmitting codewords corresponding to a single SC-LDPC chain independently, we connect consecutive chains in a multi-layer format to form a connected chain ensemble. We refer to such a transmission scheme to as continuous chain (CC) transmission of SC-LDPC codes. We show that CC transmission can be implemented with no significant increase in encoding/decoding complexity or decoding delay with respect a system using a single SC-LDPC code chain for encoding.