Abstract. Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.