  We consider generalised versions of the spin-boson model at small coupling.
We assume the spin (or atom) to sit at the origin $0 \in \mathbb{R}^d$ and the
propagation speed $v_p$ of free bosons to be constant, i.e.\ independent of
momentum. In particular, the bosons are massless. We prove detailed bounds on
the mean number of bosons contained in the ball $\{|x| \leq v_p t \}$. In
particular, we prove that, as $t \to \infty$, this number tends to an
asymptotic value that can be naturally identified as the mean number of bosons
bound to the atom in the ground state. Physically, this means that bosons that
are not bound to the atom, are travelling outwards at a speed that is not lower
than $v_p$, hence the term 'minimal velocity estimate'. Additionally, we prove
bounds on the number of emitted bosons with low momentum (soft mode bounds).
This paper is an extension of our earlier work in 'Approach to ground state and
time-independent photon bound for massless spin-boson models' (arXiv:1109.5582,
Ann. H. Poincare, 2012). Together with the results in that paper, the bounds of
the present paper suffice to prove asymptotic completeness, as we describe in
'Asymptotic completeness for the massless spin-boson model'(arXiv:1301.2357).
