Many processes in nature can be viewed as arising from subjects progressing through sequential stages and may be described by multistage models. Examples include disease development and the physiological development of plants and animals. We develop a multistage model for sampling designs where a small set of subjects is followed and the number of subjects in each stage is assessed repeatedly for a sequence of time points, but for which the subjects cannot be identified. The motivating problem is the laboratory study of developing arthropods through stage frequency data. Our model assumes that the same individuals are censused at each time, introducing among sample dependencies. This type of data often occur in laboratory studies of small arthropods but their detailed analysis has received little attention. The likelihood of the model is derived from a stochastic model of the development and mortality of the individuals in the cohort. We present an MCMC scheme targeting the posterior distribution of the times of development and times of death of individuals. This is a novel type of MCMC that uses customized proposals to explore a posterior with disconnected support arising from the fact that individual identities are unknown. The MCMC algorithm may be used for inference about parameters governing stage duration distributions and mortality rates. The method is demonstrated by fitting the development model to stage frequency data of a mealybug cohort placed on a grape vine.