  We investigate the effects of various forcings on several forms of the
Halpern-L\"auchli Theorem. For inaccessible $\kappa$, we show they are
preserved by forcings of size less than $\kappa$. Combining this with work of
Zhang in \cite{Zhang17} yields that the polarized partition relations
associated with finite products of the $\kappa$-rationals are preserved by all
forcings of size less than $\kappa$ over models satisfying the
Halpern-L\"auchli Theorem at $\kappa$. We also show that the Halpern-L\"auchli
Theorem is preserved by ${{<}\kappa}$-closed forcings assuming $\kappa$ is
measurable, following some observed reflection properties.
