Bud
The KL27 has DMA, which can be triggered by the SPI flags.
However the SPI is double-buffered so as long as you keep the buffer full you shouldn't have any gaps between bytes in a sequence.
That is, fill the next byte to be sent when the TX buffer full flag is not set and read out the received data on the RX buffer not empty flag - don't do the Tx on the Rx buffer empty flag otherwise there may be a gap due to the internal synchronisation (which is probably always there on the start byte).
Regards
Mark