Asthma prevalence has risen dramatically worldwide over the past fifty years. The cause is unknown, but a large number of risk factors have been identified. To study the causal relationships between such factors, we analyse data from a five-year longitudinal cohort study, with detailed infection and immunological data from 206 children. We employ a linear Gaussian AutoRegressive Time-Varying (ARTIVA) process to model the data, the estimation of which is done in a Bayesian framework using reversible jump Markov chain Monte Carlo. Such a model allows us to infer not only causal relationships between infection/immune phenotypes, but also when and how they change. Using this model we explore distinct stages of the developing childhood immune system. The networks reveal varying links between different infection patterns and ongoing wheeziness consistent with, but supplemental to, previous studies. They also indicate that changes in specific IgE titers predict later changes in corresponding IgG4, and that house dust mite and peanut allergy are serious risks for wheeze at five years of age, although only house dust mite allergy predicts wheeze the following year. Despite these encouraging results the linearity assumption is highly questionable because asthma development is characterised by critical values. For example, the probability of house dust mite sensitization at five years rises steadily to 93.3% for IgE titers up to 0.35 U/L and levels off for higher values. Furthermore the effect of some factors on the development of asthma may not be additive,while results involving binary (true or false) data require careful interpretation. We shall present plans to solve these problems by generalizing the existing framework to a more flexible and richer class of models. These results indicate that time-varying Bayesian networks, together with deeply phenotyped cohort studies, can be used to extract meaningful insight into the complexities of immune development.