Well, it's like fortune helped to get something or to get somewhere: a lucky accident was a transport to success))

You're right. These words have the same root -вез- . Везение (luck) is when some force or maybe Fortuna (Roman goddess of luck) takes you and "везёт" (drive) to your goal. Veine is a blood vessel or а leaf nerves. This is a stream of the life. If you get in this stream you will survive. It makes sense doesn't it?

I haven't found a good explanation about this word, there are only suppositions. The word Везение cannot be used in the sense of transportion. Пeревозка is a process of transportation, but not везение. By the way, why do French people call luck "la veine"?

If you compare these two sentences, you will understand it: Это было в среду, когда мы потерялись в глуши, и наш джип застрял после проливного дождя, но нам повезло [meaning: we got luck], и мы нашли в соседней деревне толкового мужика, который на тракторе и вывез [meaning: pulled out] нас из этой глуши / западни. So in this sentence you could try to understand relation between these words.

Ok, thank you all for your answers.