Multilabel text classification (MLTC) is an essential task in NLP applications. Traditional methods require extensive labeled data and are limited to fixed label sets. Extracting labels by LLMs is more effective and universal, but incurs high computational costs. In this work, we introduce a distillation-based T5 generalist model for zero-shot MLTC and few-shot fine-tuning. Our model accommodates variable label sets with general domain-agnostic pertaining, while modeling dependency between labels. Experiments show that our approach outperforms baselines of similar size on three few-shot tasks.