In Ireland migration was mainly from rural areas, and there was a movement of population both from rural areas to urban areas and from rural areas to other countries, of which the US was just one (though the largest one). There was substantial emigration to Britain, Canada and Australia as well.
Towns and cities did not shrink during this period; they grew. But the probably grew less, and were culturally and commercially less vibrant, than they would have done if emigration had been smaller.
Rural areas lost population. The pattern of rural settlement in Ireland was not in towns and villages, but in small hamlets and groups of dwellings called “clachans”. These disappeared, not only as a result of emigration but also as a result of official policy encouraging farmers to live on their farms rather than grouped in clachans. Farms were not often abandoned - someone else would always take over farming the land, unless it was of marginal viability - but farmhouses, both in clachans and individually on farms, were.
Many of the dwellings abandoned in the 19th century would have been packed earth structures, and would have disappeared fairly quicky in Ireland’s wet climate. However in many places traces of these settlements are still clearly visible, if you know what to look for - the outline of small field systems, large ridges indicating former areas of potato cultivation now given over to grazing, collapsed animal pens (which were often made of stone rather than packed earth, and so have lasted longer).
In the late 19th and early 20th century packed earth houses gave way to concrete, and there are many concrete dwellings which are either abandoned or used as animal sheds, feed stores and the like. Some of these were abandoned as a result of emigration, but more recently many were abandoned because the owners became more prosperous, and built themselves better homes.
The details would change, but I suspect the broad outline would be similar in other countries which experienced substantial emigration. It would be rural areas which lost population, not urban areas, and the loss would have been most visible in the most thinly-populated, marginally viable rural areas.