The influence of a thermal gradient on the propagation properties of externally excited dipolar spin waves in a magnetic insulator waveguide is investigated. It is shown that spin waves propagating towards a colder region along the magnetization direction continuously reduce their wavelength. The wavelength increase of a wave propagating into a hotter region was utilized to realize its decomposition in the partial waveguide modes which are reflected at different locations. This influence of temperature on spin-wave properties is mainly caused by a change in the saturation magnetization and yields promising opportunities for the manipulation of spin waves in spin-caloritronic applications.