Recently, researchers have proposed many multi-agent frameworks for function-level code generation, which aim to improve software development productivity by automatically generating function-level source code based on task descriptions. A typical multi-agent framework consists of Large Language Model (LLM)-based agents that are responsible for task planning, code generation, testing, debugging, etc. Studies have shown that existing multi-agent code generation frameworks perform well on ChatGPT. However, their generalizability across other foundation LLMs remains unexplored systematically. In this paper, we report an empirical study on the generalizability of four state-of-the-art multi-agent code generation frameworks across six open-source LLMs with varying parameter sizes, architectures, and performance levels. Our study reveals the unstable generalizability of existing frameworks on diverse foundation LLMs. Based on the findings obtained from the empirical study, we propose AdaCoder, a novel adaptive planning, multi-agent framework for function-level code generation. AdaCoder has two phases. Phase-1 is an initial code generation step without planning, which uses an LLM-based coding agent and a script-based testing agent to unleash LLM's native power, identify cases beyond LLM's power, and determine the errors hindering execution. Phase-2 adds a rule-based debugging agent and an LLM-based planning agent for iterative code generation with planning. Our evaluation shows that AdaCoder achieves higher generalizability on diverse LLMs. Compared to the best baseline MapCoder, AdaCoder is on average 27.69% higher in Pass@1, 16 times faster in inference, and 12 times lower in token consumption.