It depends on the extent of immigration from the home country. Mass migration of entire villages or sectors such as from Amin's Uganda in the 1970s will obviously leave areas depopulated, so won't contribute to population increase there. However most migrants continue traditional practices, such as having large families, in their adopted country.
However human nature seems to prevail when the adopted country can provide enough work - the donations to the home country kind of tail off, the youngsters see the advantages of fewer kids, and assimilation / syncretisation occurs.
Having large families may or may not be reinforced by religious belief - I'm thinking Irish migrants into England, for example, where interpretations of Catholicism led to many very large families and this persisted until quite recently, only slowy following the national UK trend towards fewer children.
People have always migrated. It's the story of humankind. It keeps the gene pool healthy and disperses ideas. It would settle down from present peak if home countries would stop being so frightful to those that live in them.