Mixing a Tagged Sample of Lead Iodide with another Iodide Salt
You are missing a basic aspect of how a precipitate in solution behaves. The precipitate isn't just statically immersed in the solution but rather some is dissolving and some of the ions (or molecules) in solution are precipitating out. So for a salt like $\ce{PbI2}$ there will be an exchange of ions which are in the precipitate with those which are in solution.