Will Shimano r440 9 speed triple shifters work with my current setup which is 9 speed 105 rear mech front mech is a triple tiagra .

yes, all you stuff is road gear.
to be honest shimano mtb shifter work with the road stuff too.

road mechs yes

ton, not quite…

9spd mtb RH shifters will work with road rear mechs
10spd mtb RH shifters won’t – different pull

MTB LH shifters won’t work with road front mechs as again they are different pul (although you can get round this if you’re running a double front mech)

my mind dont work in 10spd yet……. 😆

MTB LH shifters won’t work with road front mechs (again, different pull)

yes they will. i put deore shifters on my son’s charge plug, which is sora or tiagra.

I just edited my first response. The cable pull is different but you can bodge it on a double.

It looks like I will have to change front mech then or I might just run 1×9 as it is only for commuting .