stars always move downwards in sky when they break. although there is no gravity. Never saw falling star moves in upward direction?

A falling star is not a real star, it is a meteor.
A meteor is a bit of debris burning up from friction with our atmosphere as Earth's gravity pulls it down (of course there is gravity).

A meteor will usually appear to move downward, but depending on your location in relation to the meteor, it can appear to move upward or sideways.
But its actual motion is always down towards the planet (because of gravity).

1. "Falling stars" are not stars: they are meteors. Stars are gigantic balls of luminous plasma; meteors are caused by meteoroids which are less than 1 metre in size, usually much smaller, pea sized or like grains of sand.

2. Meteors can be seen to travel in any direction in the sky, not necessarily downward.

3. Meteors appear in the upper atmosphere of Earth, so are close enough to be affected by Earth's gravity, so many appear to travel downwards.

What is your question?

You're seeing a small meteor, not 'broken stars'. They come from space and hit our atmosphere - falling toward the ground. A few may actually reach it if they're hefty enough... but things fall down - not up.
Gravity. It's not just a good idea. It's the law.

What you call a star moving downwards and breaking is NOT a star it is a Meteorite entering the Earth's Atmosphere.

Falling Stars / Shooting Stars are old names for meteorites and not in any way connected to actual Stars.

Gravity is everywhere - it's what keeps the galaxy together, the planets orbiting our star (the Sun), the moon orbiting the Earth and us from floating away.