I've tried to create the short bursts similar to this gif but I can't seem to get the exact effect. I'm using particle world in after effects. Another problem I'm running into is the texture has some color overlay I can't get rid of.
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Generally its totally possible to achieve this with Particle World, just play around with gravity adn particle emission rate and animate them accordingly.
If you need a more advanced particle system use a plugin called Particular by Red Giant Software. Though its not free and costs money.