Changing from MTB 9 speed shifters to 10 speed road?

later 10 speed (hidden cables) 105 and above has a different pull to the previous incarnation at the rear for sure, getting the hidden cable shifters to work with 9 speed is hit and miss. MTB front mech is not the same as for a road front mech, again if you can get it to work great, but the amount the front mech moves is different due to spacing of MTB/ROAD, I know this for a fact. I also tried getting it to work on my own bike and it wasn't having any of it. 6700 STI/LX front mech.

10 speed road (shimano at least) from hidden cable variation onwards so 5700/6700/7900 is invariably NOT compatible with 9 speed mtb mechs. Yup some of the shifts will work but the pull ratio diff means invariably it'll be shit/won't work.
However what WILL work is if you get 5600/6600/7800 10 speed (so non hidden cable), these will work with a 9 speed (or 8 or even 7) mtb rear derailleur, the front though is still an issue.
Older mtb mechs like the 400/500lx I know definitely work with the triple variations of the above. Also older road mechs such as the shimano 600 ex(6207) work as both a double or triple and obviously the specific front road mech which are interchangeable on the first shimano 10 speed group and bsckwards.
The front mech is always the problem as road chainsets are spaced differently to mtb, so your road sti will pull a differing amount of cable due to the pull ratio on the front mech of your 9 speed mtb.
So road sti that are 1st gen10 speed, fine with rear, not fine with front.
Later 10 speed, you'll need to replace both mechs but give it a bash with yhe STIs with the existing mechs.
Either it'll work or you'll waste a shit ton of time trying to adjust it and never get it to work properly

believe that nearly all 10speed Shimano STI shifters have the same pull ratio as 9 speed mtb mechs - got a couple of bikes set up this way  - up until a couple of years ago if you wanted to run bigger that a 28 out back it was the only way to go - believe that Wiggins was seen in the Alps stages of the TDF with this sort of set up - exception possibly is Tiagra 4700 shifters that for some reason have an odd pull ratio 

so 10speed shimano road shifters and keep your 9 speed front and rear mech and chainset  

10 speed cassette and chain - assumes your rear hub will take a 10speed cassette OK 

useful article here http://www.bike.bikegremlin.com/2016/05/18/bicycle-rear-derailleur-compa...

ps I'm not a bike mechanic so am prepared to be corrected but I do have a couple of bikes running this type of set up - gravel and touring forums cover this issue well 

Most likely as the pull ratio is different. Look around for groupset kits with all the parts together. Chain reaction do alot of kits. Alternatively buy individual parts from multiple sources to save a couple of quid

Xx