We study continuous-time dropout in controlled differential equations. We introduce a random-batch approximation of additive vector fields. On each time interval of length $h$, a random subset of components is activated and rescaled by its inclusion probabilities, yielding an unbiased approximation of the full field. We prove a uniform-in-time mean-square trajectory error of order $\mathcal O(h)$. At the distribution level, we derive Wasserstein and pointwise density estimates, together with global $L^1$ bounds under moment assumptions. For supervised training, we prove uniform $\mathcal O(\sqrt h)$ root-mean-square fluctuations of the randomized objective and an $\mathcal O(h)$ weak error for its expectation, leading to consistency of optimal values and near-minimizers and, under quadratic growth, convergence in control space. We further characterize how the sampling law affects the error constant through inclusion probabilities and field variance, compare fixed-size sampling, fixed partitions, and Bernoulli dropout, and derive a work--accuracy model for selecting the switching scale. Numerical experiments with neural ODEs illustrate trajectory and transport errors, sampling effects, and objective consistency at a fixed control.

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