Continuous Integration (CI) configurations often need to be migrated between services (e.g., Travis CI to GitHub Actions) as projects evolve, due to changes in service capabilities, usage limits, or service deprecation. Previous studies reported that migration across CI services is a recurring need in open-source development. However, manual migration can be time-consuming and error-prone. The state-of-the-art approach, CIMig, addresses this challenge by analyzing past migration examples to create service-specific rules and produce equivalent configurations across CI services. However, its relatively low accuracy raises concerns about the overall feasibility of automated CI migration using rule-based techniques alone. Meanwhile, Large Language Models (LLMs) have demonstrated strong capabilities in code generation and transformation tasks, suggesting potential to improve the automation, usability, and generalizability of CI configuration migration. This registered report presents a study in which we aim to assess whether CI migration can be improved using LLMs. To this end, we propose CIgrate, an LLM-based framework for automatically migrating CI configurations. We plan to evaluate the performance of CIgrate compared to CIMig as a baseline, in different setups (a) zero-shot/few-shot prompting of LLMs for configuration migration and (b) fine-tuning an LLM on a dataset of already established CI service migrations. We will also seek developer feedback on the quality and usability of the generated configurations. We formulate research questions focusing on the accuracy of LLM-generated migrations versus ground truth and the output of CIMig. The expected contributions include the first LLM-powered approach for CI service migration, a comparative evaluation of its effectiveness compared to rule-based approaches, and insight into leveraging LLMs to support software configuration evolution.