Task-oriented dialogue systems are essential for applications ranging from customer service to personal assistants and are widely used across various industries. However, developing effective multi-domain systems remains a significant challenge due to the complexity of handling diverse user intents, entity types, and domain-specific knowledge across several domains. In this work, we propose DARD (Domain Assigned Response Delegation), a multi-agent conversational system capable of successfully handling multi-domain dialogs. DARD leverages domain-specific agents, orchestrated by a central dialog manager agent. Our extensive experiments compare and utilize various agent modeling approaches, combining the strengths of smaller fine-tuned models (Flan-T5-large \& Mistral-7B) with their larger counterparts, Large Language Models (LLMs) (Claude Sonnet 3.0). We provide insights into the strengths and limitations of each approach, highlighting the benefits of our multi-agent framework in terms of flexibility and composability. We evaluate DARD using the well-established MultiWOZ benchmark, achieving state-of-the-art performance by improving the dialogue inform rate by 6.6\% and the success rate by 4.1\% over the best-performing existing approaches. Additionally, we discuss various annotator discrepancies and issues within the MultiWOZ dataset and its evaluation system.