Typically, the custom breaks down in the 4th generation( I'll explain why in a second). Meaning that your great great grandchild would be Lebanese despite you being Polish.
Why? Because presumably, your son would marry a Lebanese woman, and their son/daughter a Lebanese son/daughter and so on. Thus your great great grandchild would be more Lebanese than polish due to genetics.
Consider this example. If I was chinese and moved to Italy, my grandson may look Chinese but with time, my other descendents would begin to look Italian.