A person's livelihood refers to their "means of securing the basic necessities -food, water, shelter and clothing- of life".


A general view is: It is allowed for a person living in a Muslim-majority society to emigrate to a non-Muslim society as long as it does not cause one to lose one's religion.

Beyond that, if a Muslim emigrates to the West, it is a personal decision whether or not they decide on focusing on building infrastructures in the West, or focus on going back to their country of origin. Not everyone has the same circumstances or the same role in life. Some people will work effectively in one country but not another. So it isn't the sort of thing one can say there is only one answer to. 

A more detailed view:

If someone has no choice and must emigrate for some reason, then obviously it is allowed; questions of permissibility are only for things that are by choice. Usually emigration is due to some need, such as for economic or political reasons, and not a straightforward choice.

The entire earth belongs to Allah. One of the ways that Allah directs us to one geographical location or another is through rizq - that is, making it difficult for us to live in one place and giving us employment or opportunities in another - and this is part of the divine plan.

Additionally, in reality, few (if any) Muslim-majority societies are currently embracing all the major values of Islam, such as social justice, absence of corruption, supporting the deprived, racial equality, and other things. We all know that in some Muslim-majority societies, there is also a problem with sectarian violence or sectarian restrictions.

Furthermore, in some Muslim-majority societies, Islamic practice has been restricted (such as limitations on or a bias against wearing the hijab).

So it may be overly simplistic to divide the world into "Muslim societies" and "non-Muslim societies".

However, one might surmise there are still some cultural factors in Muslim-majority societies that support a person's faith, or help in passing it on to children, such as being around mosques, seeing Islam as a normalized as part of daily life, less public alcohol consumption, and these  sorts of things. 

Anyway, yes, it seems like a good idea for Muslims to build permanent infrastructures in the West since many Muslims live in the West.

One can also note that Islam did not spread to today's "Muslim world" overnight. One major reason for the spread of Islam was the migration of individual Muslims to various places, and their their establishment of mosques and other institutions. So building Islamic institutions in the West is not something new or different, rather, it is just the same sort of thing that happened before. Sometimes there is a mental image that the Muslim-majority world was always that way, but that is obviously not the case on a historical level.

Note: There is an underlying assumption between this statement that all Muslims in the West come from, or have ancestry in, other countries. This assumption should be challenged. Not all Muslims in the West were born in other countries. Many Muslims in the West were born in the West and cannot easily return to the country of their ancestors, or they may be of mixed ancestry and not have a specific country to go do that is "theirs".

Also, not all Muslims in the West trace their ancestry to non-Western countries, so they do not have another country to go back to. (While some Western Muslims have attempted "hijra" to the Muslim-majority world, there are usually barriers along the lines of residency permits, work permits, being treated perpetually like an outsider, etc.)

I suspect that the migration of Muslims to the West is part of Allah's plan and perhaps Allah wishes to end the "clash of civilizations" between Islam and the West and create more interchange. Also, having a significant number of Muslims in the West has sparked some important discussions about Islam, Islamic law, inter-faith relations, and other matters due to the situation of Islam being in a new environment. This helps to have growth in Muslim thought rather than stagnation. So one can see there have been some benefits overall for the ummah to having a large number of Muslims, and Muslim institutions, in the West.