We compute the connected components of arbitrary parahoric level affine Deligne-Lusztig varieties for quasisplit reductive groups, by relating them to the connected components of infinite level moduli spaces of p-adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites.As an application, we deduce new CM lifting results on integral models of Shimura varieties at arbitrary parahoric levels. We also prove various results beyond the quasi-split case, by studying the cohomological dimensions of Newton strata inside the B + dR -affine Grassmannian. Contents 1. Introduction 1 Acknowledgements 10 2. Preliminaries and background 10 3. Generic Mumford-Tate groups 28 4. Proof of main theorems 31 5. Beyond the quasisplit case. 40 References 45
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