  We use the latest multi-redshift ($z=6.5-8.7$) upper limits on the 21-cm
signal from the Murchison Widefield Array (MWA) to explore astrophysical models
which are inconsistent with the data. These upper limits are achieved using 298
h of carefully excised data over four observing seasons. To explore these upper
limits in the context of reionisation astrophysics, we use 21CMMC. We then
connect the disfavoured regions of parameter space to existing observational
constraints on reionisation such as high-$z$ galaxy ultra-violet (UV)
luminosity functions, background UV photoionisation rate, intergalactic medium
(IGM) neutral fraction, the electron scattering optical depth and the soft-band
X-ray emissivity. We find the vast majority of models disfavoured by the MWA
limits are already inconsistent with existing observational constraints. These
inconsistent models arise from two classes of models: (i) `cold' reionisation
and (ii) pure matter density fluctuations (i.e. no reionisation). However, a
small subsample of models are consistent implying the existing MWA limits
provide unique information in disfavouring models of reionisation, albeit
extremely weakly. We also provide the first limits on the soft-band X-ray
emissivity from galaxies at high redshifts, finding $1\sigma$ lower limits of
$\epsilon_{{\rm X},0.5-2~{\rm keV}}\gtrsim10^{34.5}$ erg s$^{-1}$ Mpc$^{-3}$.
Finally, we recover 95 per cent disfavoured limits on the IGM spin temperature
of $\bar{T}_{\rm S}\lesssim$ 1.3, 1.4, 1.5, 1.8, 2.1, 2.4 K at $z=6.5, 6.8,
7.1, 7.8, 8.2, 8.7$. With this we infer the IGM must have undergone, at the
very least, a small amount of X-ray heating. Note, the limits on
$\epsilon_{{\rm X},0.5-2~{\rm keV}}$ and $\bar{T}_{\rm S}$ are conditional on
the IGM neutral fraction.
