  We evaluate the QED corrections due to initial state radiation (ISR) to
associated Higgs boson production in electron-positron ($e^+ e^-$) annihilation
at typical energies of interest for the measurement of the Higgs properties at
future $e^+ e^-$ colliders, such as CEPC and FCC-ee. We apply the QED Structure
Function approach to the four-fermion production process $e^+ e^- \to \mu^+
\mu^- b \bar b$, including both signal and background contributions. We
emphasize the relevance of the ISR corrections particularly near threshold and
show that finite third order collinear contributions are mandatory to meet the
expected experimental accuracy. We analyze in turn the r\^ole played by a full
four-fermion calculation and beam energy spread in precision calculations for
Higgs physics at future $e^+ e^-$ colliders.
