The phrase "rapid influx" means a sudden arrival or increase of something or someone, usually in large numbers, happening very quickly. Full definition
In some places the way in which we saw rapid influx of in particular eastern European migrants after the accession of those countries to Europe definitely caused downward pressure on wages, definitely caused changes to local terms and conditions to some workers in some sectors.» (politics.co.uk)
That change is being spurred along by the rapid influx of new millionaires in America, over 400,000 in 2016. (inc.com)
The rapid influx of Jewish refugees into Palestine has resulted in many cases in the rise of a quasi-elite who have not been able to provide true leadership for the communes and have come into conflict with the genuine chaluzim. (religion-online.org)