The combination of large neighbourhood search (LNS) methods with complete search methods has proved to be very effective. By restricting the search to (small) areas around an existing solution, the complete method is often able to quickly improve its solutions. However, developing such a combined method can be time-consuming: While the model of a problem can be expressed in a high-level solver-independent language, the LNS search strategies typically need to be implemented in the search language of the target constraint solvers. In this paper we show how we can simplify this process by (a) extending constraint modelling languages to support solver-independent LNS search definitions, and (b) defining small solver extensions that allow solvers to implement these solver-independent LNS searches. Modellers can then implement an LNS search to be executed in any extended solver, by simply using the modelling language constructs. Experiments show that the resulting LNS searches only introduce a small overhead compared to direct implementations in the search language of the underlying solvers.