Car stalls and putts when accelerating and braking

I would take it to a dealer or good shop and have them run a scan to see if there are powertrain codes stored. Not all codes will turn on the light, some will be there but unless scanned they are not discovered. If there are no codes stored or pending then the job becomes more difficult.

First question - What engine does it actually have and gas or diesel. You have 3 cyl but they used a 2.0 and 2.4 gas and a 2.0 diesel in EU and Aus. Also which transmission

What codes were found when it was scanned if any? With that description I would expect a code or two. If there are no codes then you will need to attach a scan tool on it and watch the live data to see what is dropping out or changing. There are multiple parts that could cause the engine to act this way, anything from a bad crankshaft sensor, failing O2 sensor, loose battery cable, bad engine ground or even transmission problems if it has the CVT unit as just a few examples. However without reading the data you could replace every part and still miss the actual cause. In the auto world we call that "firing the parts cannon" and it is very expensive. We try not to do that if at all possible.

Here are a few questions that might help narrow it down.

Did this just start doing this?
Were ANY changes made before it started (new parts, plugs, sensors, different gas station)?
Does it seem to happen more when the engine is cold or does it need to warm up?
Does it seem to be better or worse if it rains/snows?
Does outside temperature effect it?
You say it does it while accelerating, if it starts acting up and you let off the fuel what happens?
Does it seem to be different if you slowly accelerate or if you just stomp the pedal down?