Any help would be much appreciated

If you are not using Oneshot, "Enable PWM input" needs to be ON. Otherwise, the ESCs expect a Oneshot signal. It is switched off by default so that ESCs don't confuse the normal PWM input with Oneshot signals. Also, you can use damped light even if the ESCs don't support Oneshot as they are not related. Damped light is just active braking of the motors. The ESCs should support damped light though as it puts a bit more stress on them.

few things you could try.
in blheli u can change demag to low from high.
and enable pwm input if not using oneshot.
calibrate using motor tab after that. 

Try to swap motors and check whether its the problem of ESC i think it stupid to say but it just for confirmation.Try the motor rotating good with ESC that has bad motor which jerks a little bit. This is because of programming malfunction inside ESC i faced when i tried my firmware with ESC's and motors .That may be sometimes due to motors unable to catch the speed of ESC's due to motor damages. i think try swapping motors.

I did following things
Demag to HIGH
PWM Input to ON
And then calibrated

Still same

Try low demag, and medium-high timing. Also, try posting on the BLHeli thread for more help.

EDIT: Could you please post a link to the ESCs? I want to see what processor they are using and the other specs.

EDIT 2: What is the cycle time in Cleanflight?

Demag does not helps.

Link to ESC: http://www.emaxmodel.com/esc/nano-series-esc/emax-nano-series-esc-12a.html

Cycle time? where do i find that?

That ESC appears to support Oneshot125. Maybe you could try switching on Oneshot in the configuration menu of Cleanflight and setting "Enable PWM input" to off. It also is supposed to have a SiLabs F330 so it should be able to handle these motors, especially on BLHeli 14.4. The cycle time can be found in the bottom left corner of CF configurator. If the looptime is low, normal PWM input to the ESC is not enough, which is why we use Oneshot.

Sathya ji, i have some doubts regarding this, may be foolish

1) please check anything wrong with ESCs Ground connection.

2) any other devices connected in Signal pin of the ESC like configuration device? if yes then remove other devices and try.

3) also try giving minimum speed to spin first (just like spin when armed parameter) and give full throttle.

@SK they say that it is supported but as the RCGroups they say that it does not
http://www.rcgroups.com/forums/showpost.php?p=32964164&postcount=48

And I get the same issue prior to flashing the factory BLHeli firmware which is not basically BLHELI but it is based on the BLHeli (playing with words)

But after I flashed with the normal BLHeli, I don't see any difference. With oneShot or without OneShot I don't see any difference they show the same problem

@Akhil

How do i check point 1?
2. Nothing is connected on the signal pin I am sure. I have directly soldered the signal wire from the FC to ESC
3. When I slowly give throttle it works fine, but if I instantly change the throttle it gives the problem.

Which few test with slow throttle response it flew nicely but if I give quick throttle response it shutters or entirely stops which makes the quad to fall down.

After reading through that thread again, I have a couple of suggestions:

• Connect the ground wire of the ESC signal cable to the FC as well. Connecting only the white signal wire can cause strange behaviour.
• Check whether this is a V1 or V2 ESC. The V1 appears to use the XP-12A hex that you flashed and does not support damped light. V2 uses the Emax_Nano_20A hex in BLHeli. Once you flash normal BLHeli on it Oneshot should work but that doesn't seem to be related to the problem

I have attached a screenshot from the BLHeli manual. If your ESC looks like this it is the V2 and needs the Emax_Nano_20A hex.

I also wanted to ask how you are flashing and connecting to the ESCs. Is it over the C2 interface (the pads on the PCB)? I can't tell for sure from the video.

This is V1, so its XP12A (i did read the thread and found this )

The mini Flip32 does not provide option to connect ground

http://www.aliexpress.com/item/Super-Mini-Flip32-10-DOF-Flight-Controller-Board-with-shell-For-QAV250-ZMR250-mini-150-180/32577256196.html

Mini Flip32+ takes only CPPM input from RX and only 1 cable per esc (signal)

I flashed using the C2 interface the V1 is very bad design by the emax and Suppliers dont have V2 at the moment.

Take GND from rx connection, did you tried by connecting the ESCs directly to RX without FC using pwm?

also try reducing Startup power.

Good point Akhil I will try that.

I will do both the points and get back to you.