Globular clusters (GCs) are key to understanding the formation and evolution of our Galaxy. While the abundances of light and Fe-peak elements in GCs have been widely studied, investigations into heavier, neutron-capture elements -- and their connection to multiple stellar populations and GC origins -- remain limited. In this work, we analysed the chemical abundances of neutron-capture elements in GCs to trace the Galactic halo and to explore possible links to the MP phenomenon. Our goal is to better constrain the nature of the polluters responsible for intracluster enrichment and to distinguish the origin of GCs through the chemical signature of neutron-capture elements. We examined 14 GCs from the Gaia-ESO Survey, spanning a wide metallicity range, [Fe/H] from -0.40 to -2.32, using a homogeneous methodology. We focused on the abundances of Y, Zr, Ba, La, Ce, Nd, Pr, and Eu, derived from FLAMES-UVES spectra. These were compared with predictions from a stochastic Galactic chemical evolution model. With the exception of Zr, the model broadly reproduces the observed trends in neutron-capture elements. In some GCs, we found strong correlations between hot H-burning products (Na, Al) and s-process elements, pointing to a shared nucleosynthesis site, e.g., asymptotic giant branch stars of different masses and/or fast-rotating massive stars. We also detect a distinct difference in [Eu/Mg] ratio between in-situ (<[Eu/Mg]>=0.14dex) and ex-situ (<[Eu/Mg]>=0.32dex) GCs, highlighting their different enrichment histories. Finally, on average, Type II GCs (NGC 362, NGC 1261, and NGC 1851) showed a s-process element spread ratio between second- and first-generations about twice as large as those seen in Type I clusters.