We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes also known as testing by betting, as well as tests based on combining p-values. Simulation and empirical studies are conducted for a few settings of the null hypotheses, and they show that methods based on e-processes are efficient.

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