In your master report, put your subreports into block-level bands. Block-level elements span the whole width of the page and thus make it easy for the engine to make breaks there.
Assuming that everything into the report-header, then make your report-header a block-layout band, and add subbands. Each of these subbands then can have pagebreaks and they will be honoured by the engine.
This can also be done without subband directly giving pagebreaks to the subreports from the Attributes Pane.