Due to its dire impacts on marine life, public health, and socio-economic services, oil spills require an immediate response. Effective action starts with good knowledge of the ocean dynamics and circulation, from which Lagrangian methods derive key information on the dispersal pathways present in the contaminated region. However, precise assessments of the capacity of Lagrangian methods in real contamination cases remain rare and limited to large slicks spanning several hundreds of km. Here we address this knowledge gap and consider two medium-scale (tens of km wide) events of oil in contrasting conditions: an offshore case (East China Sea, 2018) and a recent near-coastal one (East Mediterranean, 2021). Our comparison between oil slicks and Lagrangian diagnostics derived from near-real-time velocity fields shows that the calculation of Lagrangian fronts is, in general, more robust to errors in the velocity fields and more informative on the dispersion pathways than the direct advection of a numerical tracer. The inclusion of the effect of wind is also found to be essential, being capable of suddenly breaking Lagrangian transport barriers. Finally, we show that a usually neglected Lagrangian quantity, the Lyapunov vector, can be exploited to predict the front drifting speed, and in turn, its future location over a few days, on the basis of near-real-time information alone. These results may be of special relevance in the context of next-generation altimetry missions that are expected to provide highly resolved and precise near-real-time velocity fields for both open ocean and coastal regions.