Numerous eruptive variable young stellar objects (YSOs), mostly Class I systems, were recently detected by the near-infrared Vista Variables in the Via Lactea (VVV) survey. We present an exploratory near-infrared spectroscopic variability study of 14 eruptive YSOs. The variations were sampled over 1-day and 1 to 2-year intervals and analysed in combination with VVV light curves. CO overtone absorption features are observed on 3 objects with FUor-like spectra: all show deeper absorption when they are brighter. This implies stronger emission from the circumstellar disc with a steeper vertical temperature gradient when the accretion rate is higher. This confirms the nature of fast VVV FUor-like events, in line with the accepted picture for classical FUors. The absence of Br$gamma$ emission in a FUor-like object declining to pre-outburst brightness suggests that reconstruction of the stellar magnetic field is a slow process. Within the 1-day timescale, 60% of H$_2$-emitting YSOs show significant but modest variation, and 2/6 sources have large variations in Br$gamma$. Over year-long timescales, H$_2$ flux variations remain modest despite up to 1.8 mag variation in $K_s$. This indicates that emission from the molecular outflow usually arises further from the protostar and is unaffected by relatively large changes in accretion rate on year-long timescales. Two objects show signs of on/off magnetospheric accretion traced by Br$gamma$ emission. In addition, a 60% inter-night brightening of the H$_2$ outflow is detected in one YSO.
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