Turkey is on the road to becoming EU Member.

Can You tell me the grammatical explanation of 'to becoming' ,,

Can you give me more examples about ------to + Verbing --------

I just knew the---looking forward to + Verbing ---- form.
I will only comment on the statement, 'Turkey is on the road to becoming EU Member', because I think it is a little bit incorrect semantically.

I believe it should be written, 'Turkey is on the road to being an EU member.'

Maybe it's the 'road of becoming', and 'the road to being'. The name of the road is 'Becoming-a-Member Road'. The road leads to 'Being a Member of the EU'.

(I don't think this helps you otherwise. Someone else will step in on that.)
The 'to' in the expression 'on the road to...' is a preposition. Therefore, it requires a noun(including a gerund) as an object. The 'road' in your example was used in a figurative sense. The word can, of course, be used to refer to a real road. In whichever sense you may use it, the road is something that leads to a certain thing or place. The 'to' in your example has the connotation of 'the road that leads to a thing or place.'

If a car pulls up and its driver asks you which way he should go to get to Istanbul, you may reply,"You are on the right road to Istanbul."

Other examples:

a gateway to becoming a teacher

the stairs to the rooftop

the door to the basement

the hallway to the gym

Here are more:

be used to -ing

be accustomed to -ing

be dedicated to -ing

be devoted to -ing

What do you say to -ing?

be opposed to -ing

object to -ing

with a view to -ing

Hope this helps.