Johnny Holland 1 Try, 2 Conversions
CJ Stander 1 Try
Darren Sweetnam 1 Try
Gerhard van den Heever 1 Try
Plymouth Albion's bonus-point win against Nottingham meant that Munster A needed to win by 37 points to secure top spot in the pool but they fell 11 points short and go into the quarters as sixth seeds.
Munster A took an early lead through Ronan O'Mahony's sixth try in six B&I Cup games with academy out-half Johnny Holland adding the extras to make it 7-0 after nine minutes.
Despite that early score, Munster A were unable to extend the lead and the half ended 7-0.
They upped the tempo in the second half and Johnny Holland crossed for the second try after 47 minutes before adding the conversion himself to make it 14-0.
Holland had to be withdrawn with an injury shortly after but it didn't halt his side's momentum and CJ Stander crossed for the third try after 50 minutes with Gerry Hurley kicking the conversion to make it 21-0.
Darren Sweetnam touched down the bonus-point try after 57 minutes to put the win beyond doubt with Rory Scannell, Holland's replacement, narrowly off target with the conversion.
Gerhard van den Heever scored Munster's fifth, his third try in eight days, to make it 31-0 with Scannell again unlucky with the conversion.
Fraser Lyle touched down Stirling's only score after 65 minutes to reduce the arrears but it was only a consolation as Munster A secured their passage to the last eight.
Brian Archibald was sin-binned for a deliberate knock-on late in the game with Munster A substitute Brian Haugh following him to the bin in the final minute but it made no difference to the final 31-5 scoreline.
Academy prop Niall Horan came on for Alan Cotter to make his British and Irish Cup debut after 74 minutes.
Stirling County: Ben Addison; Graham Lindsay, Danny Gilmour, Brian Archibald, Rory Hughes; Stuart Edwards, Sean Kennedy; Jamie Bhatti, Alex Moffat (Capt.), Garry Mountford; Craig Deacons, Perry Parker, Dimitri Basilia, Shaun McDonald, Jamie Swanson. Replacements: Reyner Kennedy, Kris Burney, Colin Hutton, Euan Dewar, Will Bordill, AJ McFarlane, Fraser Lyle.
Munster A: Ronan O'Mahony; Gerhard van den Heever, Cian Bohane, Denis Hurley, Darren Sweetnam (Niall Kenneally 66); Johnny Holland (Rory Scannell 50), Gerry Hurley (Brian Haugh 53); Alan Cotter (Niall Horan 74), Niall Scannell (Ger Slattery 74), John Ryan; Ian Nagle, Billy Holland (Darren O'Shea 74); Dave O'Callaghan (Shane Buckley 53), Barry O'Mahony (Capt.), CJ Stander. Replacements: Ger Slattery, Niall Horan, Darren O'Shea, Shane Buckley, Brian Haugh, Rory Scannell, Niall Kenneally.