  Massive black hole binaries (MBHBs) of $10^5 \, \rm M_\odot - 3 \times 10^7
\, \rm M_\odot $ merging in low redshift galaxies ($z\le4$) are sufficiently
loud to be detected weeks before coalescence with the Laser Interferometer
Space Antenna (LISA). This allows us to perform the parameter estimation $on$
$the$ $fly$, i.e. as a function of the time to coalescence during the inspiral
phase, relevant for early warning of the planned LISA protected periods and for
searches of electromagnetic signals. In this work, we study the evolution of
the sky position, luminosity distance, chirp mass and mass ratio uncertainties
as function of time left before merger. Overall, light systems with total
intrinsic mass $\rm M_{\rm tot} = 3 \times 10^5 \, \rm M_\odot$ are
characterized by smaller uncertainties than heavy ones ($\rm M_{\rm tot} = 10^7
\, \rm M_\odot$) during the inspiral. Luminosity distance, chirp mass and mass
ratio are well constrained at the end of the inspiral. Concerning sky position,
at $z=1$, MBHBs with $\rm M_{\rm tot} = 3 \times 10^5 \, \rm M_\odot$ can be
localized with a median precision of $\simeq 10^2 \, \rm deg^2 (\simeq 1 \, \rm
deg^2)$ at 1 month (1 hour) from merger, while the sky position of heavy MBHBs
can be determined to $10 \, \rm deg^2$ only 1 hour before merger. However the
uncertainty around the median values broadens with time, ranging in between
0.04 -- 20 $\rm deg^2$ (0.3 -- 3 $\times 10^3 \, \rm deg^2$) for light (heavy)
systems at 1 hour before merger. At merger the sky localization improves down
to $\simeq 10^{-1} \, \rm deg^2$ for all masses. For the benefit of the
observer community, we provide the full set of data from our simulations and
simple and ready-to-use analytical fits to describe the time evolution of
uncertainties in the aforementioned parameters, valid for systems with total
mass between $10^5$--$10^7 \, \rm M_\odot$ and redshift $0.3$--$3$.
