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Modern network datasets are often composed of multiple layers, either as different views, time-varying observations, or independent sample units, resulting in collections of networks over the same set of vertices but with potentially different connectivity patterns on each network. These data require models and methods that are flexible enough to capture local and global differences across the networks, while at the same time being parsimonious and tractable to yield computationally efficient and theoretically sound solutions that are capable of aggregating information across the networks. This paper considers the multilayer degree-corrected stochastic blockmodel, where a collection of networks share the same community structure, but degree-corrections and block connection probability matrices are permitted to be different. We establish the identifiability of this model and propose a spectral clustering algorithm for community detection in this setting. Our theoretical results demonstrate that the misclustering error rate of the algorithm improves exponentially with multiple network realizations, even in the presence of significant layer heterogeneity with respect to degree corrections, signal strength, and spectral properties of the block connection probability matrices. Simulation studies show that this approach improves on existing multilayer community detection methods in this challenging regime. Furthermore, in a case study of US airport data through January 2016 -- September 2021, we find that this methodology identifies meaningful community structure and trends in airport popularity influenced by pandemic impacts on travel.