Tunnelling is one of the most fundamental manifestations of quantum mechanics. The recent advent of lightwave-driven scanning tunnelling microscopy has revolutionized ultrafast nanoscience by directly resolving electron tunnelling in electrically conducting samples on the relevant ultrashort length- and timescales. Here, we introduce a complementary approach based on terahertz near-field microscopy to perform ultrafast nano-videography of tunnelling processes even in insulators. The central idea is to probe the evolution of the local polarizability of electron-hole pairs with evanescent terahertz fields, which we detect with subcycle temporal resolution. In a proof of concept, we resolve femtosecond interlayer transport in van der Waals heterobilayers and reveal pronounced variations of the local formation and annihilation of interlayer excitons on deeply subwavelength, nanometre scales. Such contact-free nanoscopy of tunnelling-induced dynamics should be universally applicable to conducting and non-conducting samples and reveal how ultrafast transport processes shape functionalities in a wide range of condensed matter systems. Subcycle nano-videography of charge-transfer dynamics in WSe2/WS2 heterostructures is obtained by using a terahertz near-field microscopy. The central idea is to probe the local polarizability of electron-hole pairs with evanescent terahertz fields.