Theoretically by the book the answer is "YES". But in real life the anwser is "not necessary". If your Bazi and luck is excellent, staying put does not necessary means bad. If you Bazi is bad, go and work at some place or region which according to your Bazi is good, does not promise your prosperity.
Again take Tan Sri Dato Seri (Dr.) Lim Goh Tong 林梧桐 bazi life chart. Fire is his unfavorable elements which South represent Fire.Born in 1918 in Anxi, and as China's situation was becoming worse, he decided to take the plunge in venturing out to Malaysia (known as Malaya at that time).
Moving to Malaysia from China means that he moved SOUTH which represent FIRE.
He worked for his uncle as a carpenter in the first two years, learning the Malay language in the meantime. After gaining experience he went on to become a building sub-contractor and completed his first job of building a two-storey school. He returned to Anxi in 1940, but went back to Malaya the next year amidst a deteriorating war environment.
Again you noticed that he moved SOUTH which represent FIRE.
Again Malaya in Chinese 马来西亚 has the word "Horse" pronounced as "MA 马" which represented " 午 or WU" which again represent FIRE.
He was a prominent wealthy Malaysian Chinese businessman, he was renowned for his vision and courage in transforming Genting Highlands
from an unexplored hilltop into one of the world’s most successful casino resorts. He was once the richest man in Malaysia.
Now, you should be able to catch what I trying to said.
Your other question-Does our name play an important rule with our luck?
Cheers.
Peter Yap