2003 406 HDi Estate Automatic

Hi

Never owned a Peugeot before so don't know what to expect from the automatic box., or the car for that matter. I recently test drove a 2003 HDi 110 auto estate with comprehensive independent service history. It's clearly been looked after and is clean and tidy throughout. It seems to be a bit of sluggish drive though and although it changes through the gears cleanly and smoothly, it does not appear to kick-down from 4th to 3rd. If I manually down shift, it revs at 4k but not beyond and when I manually up-shift it takes a few seconds before it goes back into 4th. There are no diagnostic lights on the dash and the sport and winter modes seem to activate when pressed and are lit on the dash to confirm. There is nothing in the service history to show any gearbox work or fluid changes. The car has done 113000 miles and aside from the gearbox anomaly appears to be a sweet drive...just a bit slow. Are these gearbox symptoms a likely need of new solenoids or is this just how the automatics are. Is it possible that a software upgrade and resetting of adaptive driving might help? Any thoughts or advice welcome.

thank you