When counting continuously moving individuals over survey regions across time, the statistical properties of these counts are intimately connected with the underlying movement. In a recent work, the multivariate distribution of this kind of counts has been studied in detail, showing that if the movement is modelled with a continuous-time stochastic process, the space-time intensity and covariance of the count field have a special structure which can be used to infer movement parameters from count data. Here we show that the same distribution can be applied to the case with binomial detection error, providing thus a more realistic model for real abundance data found in ecological applications. The models can be understood as continuous evolutions of binomial or Poisson spatial point processes, and allow, in theory, to split population size rates with detection probabilities. Since the explicit likelihoods are currently intractable, composite likelihood methods are here used for inference. We present two illustrations for ecological data. The first is on an experimental data set on fruit flies being released on a meadow and subsequently captured in order to quantify dispersal. The movement of the flies is modelled with an Itô difussion with temporally varying advection and diffusivity, both being linked to daylight, temperature and wind through experimentally-grounded models. The second is on citizen-science alike data on british red kites during their 2009 breeding season, where parameters of an overall model considering nest positioning, home range, mixture of floating and breeding population, detection probability and behaviour around a feeding center must be estimated.