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.