We study two one-dimensional variants of the contact process: the contact-and-barrier process, where the population evolves in a region delimited by a randomly moving barrier, and the multitype contact process, in which two species compete for space. The contact-and-barrier process is started with the barrier at the origin and all sites to its right occupied, while the multitype contact process is started from the Heaviside configuration with species 1 to the left of the origin and species 2 to the right. We prove that both models exhibit tight interfaces and that, after centring by an appropriate deterministic speed, the interface position satisfies a central limit theorem. Our analysis relies on a renewal-time method based on a novel construction called patchwork construction, in which the processes are built by concatenating space-time evolutions over successive time intervals of random length, providing a more convenient framework for defining the renewal times that drive the proofs.