  We addressed the so far unexplored issue of outflows induced by exponentially
growing power sources, focusing on early supermassive black holes (BHs). We
assumed that these objects grow to $10^9\;M_{\odot}$ by z=6 by
Eddington-limited accretion and convert 5% of their bolometric output into a
wind. We first considered the case of energy-driven and momentum-driven
outflows expanding in a region where the gas and total mass densities are
uniform and equal to the average values in the Universe at $z>6$. We derived
analytic solutions for the evolution of the outflow, finding that, for an
exponentially growing power with e-folding time $t_{Sal}$, the late time
expansion of the outflow radius is also exponential, with e-folding time of
$5t_{Sal}$ and $4t_{Sal}$ in the energy-driven and momentum-driven limit,
respectively.
  We then considered energy-driven outflows produced by QSOs at the center of
early dark matter halos of different masses and powered by BHs growing from
different seeds. We followed the evolution of the source power and of the gas
and dark matter density profiles in the halos from the beginning of the
accretion until $z=6$. The final bubble radius and velocity do not depend on
the seed BH mass but are instead smaller for larger halo masses. At z=6, bubble
radii in the range 50-180 kpc and velocities in the range 400-1000 km s$^{-1}$
are expected for QSOs hosted by halos in the mass range
$3\times10^{11}-10^{13}\;M_{\odot}$.
  By the time the QSO is observed, we found that the total thermal energy
injected within the bubble in the case of an energy-driven outflow is
$E_{th}\sim5 \times 10^{60}$ erg. This is in excellent agreement with the value
of $E_{th}=(6.2\pm 1.7)\times 10^{60}$ erg measured through the detection of
the thermal Sunyaev-Zeldovich effect around a large population of luminous QSOs
at lower redshift. [abridged]
