While Speech Large Language Models (Speech-LLMs) have achieved strong performance on adult Automatic Speech Recognition (ASR), their effectiveness on child speech remains under-explored, and single models often struggle to handle diverse adult and child age groups simultaneously. This paper proposes a Mixture-of-Experts (MoE) Speech-LLM for unified ASR across adult and child speech spanning diverse environments and age groups. The framework employs a Classifier-based Domain Router (C-DR) with a coarse-to-fine strategy and integrates both a Mixture-of-Projectors (MoP) and a Mixture-of-LoRAs (MoL) to model domain-specific variations. To address routing uncertainty near domain boundaries, an Entropy-Aware Routing (EAR) mechanism is introduced to dynamically incorporate a shared expert. Experiments on public child corpora demonstrate consistent improvements over baselines while preserving adult ASR performance. To our knowledge, this is the first work leveraging Speech-LLMs for unified, multi-domain ASR encompassing both children and adults.