SUSY models with compressed mass spectra are highly motivated by theoretical considerations, such as dark matter constraints and naturalness. The presented search targets simplified models of SUSY in which the signal consists of top squark (stop) pair production, followed by four-body decays or decays via an intermediate chargino. The models assume the neutralino to be the lightest supersymmetric particle (LSP), with a mass gap between the stop and the LSP that is smaller than the W boson mass. The difficulty with identifying the resulting soft decay products can be mitigated by requiring the presence of an initial-state radiation jet, which boosts the system. The data is analysed using two approaches: a sequential selection based on discriminating variables and a multivariate technique. The results are based on a sample of proton-proton data from Run II of the Large Hadron Collider, recorded with the CMS detector at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of ${35.9~\mathrm{fb}^{-1}}$. No significant deviation from the standard model is observed and upper limits are set on the stop production cross section in the plane of the top squark vs. LSP masses, excluding stop masses up to 560 GeV. Furthermore, a statistical combination with the results of an all-hadronic channel is presented, yielding the most stringent limits in this region of the SUSY parameter space with 2016 data.