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Dialogue summarization task involves summarizing long conversations while preserving the most salient information. Real-life dialogues often involve naturally occurring variations (e.g., repetitions, hesitations) and existing dialogue summarization models suffer from performance drop on such conversations. In this study, we systematically investigate the impact of such variations on state-of-the-art dialogue summarization models using publicly available datasets. To simulate real-life variations, we introduce two types of perturbations: utterance-level perturbations that modify individual utterances with errors and language variations, and dialogue-level perturbations that add non-informative exchanges (e.g., repetitions, greetings). We conduct our analysis along three dimensions of robustness: consistency, saliency, and faithfulness, which capture different aspects of the summarization model's performance. We find that both fine-tuned and instruction-tuned models are affected by input variations, with the latter being more susceptible, particularly to dialogue-level perturbations. We also validate our findings via human evaluation. Finally, we investigate if the robustness of fine-tuned models can be improved by training them with a fraction of perturbed data and observe that this approach is insufficient to address robustness challenges with current models and thus warrants a more thorough investigation to identify better solutions. Overall, our work highlights robustness challenges in dialogue summarization and provides insights for future research.