This paper presents Athanor, a novel local search solver that operates on abstract constraint specifications of combinatorial problems in the Essence language. It is unique in that it operates directly on the high level, nested types in Essence, such as set of partitions or multiset of sequences, without refining such types into low level representations. This approach has two main advantages. First, the structure present in the high level types allows high quality neighbourhoods for local search to be automatically derived. Second, it allows Athanor to scale much better than solvers that operate on the equivalent, but much larger, low-level representations. The paper details how Athanor operates, covering incremental evaluation, dynamic unrolling of quantified expressions and neighbourhood construction. A series of case studies show the performance of Athanor, benchmarked against several local search solvers on a range of problem classes.