  The main goal of the paper is to prove convergence in norm and pointwise
almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter
polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous
flows. We will pay special attention to quantitative aspects of pointwise
convergence phenomena from the point of view of uniform oscillation estimates
for multiparameter polynomial Radon averaging operators. In the proof of our
main result we develop flexible Fourier methods that exhibit and handle the
so-called "parameters-gluing'' phenomenon, an obstruction that arises in
studying oscillation and variation inequalities for multiparameter polynomial
Radon operators. We will also discuss connections of our main result with a
multiparameter variant of the Bellow-Furstenberg problem.
