  Federated learning (FL) can suffer from a communication bottleneck when
deployed in mobile networks, limiting participating clients and deterring FL
convergence. The impact of practical air interfaces with discrete modulations
on FL has not previously been studied in depth. This paper proposes a new
paradigm of flexible aggregation-based FL (F$^2$L) over orthogonal frequency
division multiple-access (OFDMA) air interface, termed as ``OFDMA-F$^2$L'',
allowing selected clients to train local models for various numbers of
iterations before uploading the models in each aggregation round. We optimize
the selections of clients, subchannels and modulations, adapting to channel
conditions and computing powers. Specifically, we derive an upper bound on the
optimality gap of OFDMA-F$^2$L capturing the impact of the selections, and show
that the upper bound is minimized by maximizing the weighted sum rate of the
clients per aggregation round. A Lagrange-dual based method is developed to
solve this challenging mixed integer program of weighted sum rate maximization,
revealing that a ``winner-takes-all'' policy provides the almost surely optimal
client, subchannel, and modulation selections. Experiments on multilayer
perceptrons and convolutional neural networks show that OFDMA-F$^2$L with
optimal selections can significantly improve the training convergence and
accuracy, e.g., by about 18\% and 5\%, compared to potential alternatives.
