In many countries in the region (like Hungary, Serbia and Czechia) the surname Horvat or similar is very common. Was there ever a large scale migration from Croatia to this region? Is there any other reason why this particular surname is so common?
I can offer an insight from the point of view of the Hungarian Kingdom.
I'd like to start first by pointing out that "Horvat" is not the only such common surname in Hungary: Horvát(h) - Croatian, Rác(z) - Serbian, Oláh - Romanian, Német(h) - German, Tót(h) - Slovakian. These are all very common in Hungary, so in this sense migrational movements were not restricted only to Croatians.
To answer the question, were there any mass migrations from Croatia, the answer is yes. While this region has quite a turbulent history for the past 1000 years, there are two main periods in the Kingdom of Hungary which are relevant. First, the brief, but brutal Mongol invasion in 1241-42, and second, the 150-year-long Turkish occupation and endless back-and-forth war from 1526-1686. Both of these conflicts left the Kingdom in a devastated state, not only in terms of infrastructure but of manpower: estimates vary, but both events caused a loss of around 10-30% of the population.
Naturally, the rulers of the Kingdom (first the Hungarian House of Árpád, later the Austrian House of Habsburg) invited in settlers from the neighbouring countries and lands. This was beneficial for both parties: the settlers/immigrants received good opportunities (such as land and a stable state) to start a new life, while the Kingdom itself got to keep running. Remember, back then nationality was not that important. What mattered is you paid your taxes and answered the call for war if needed.
The immigrations included all the ethnicities I mentioned above, including Croatians from the Kingdom of Croatia (technically the Kingdom of Croatia and Kingdom of Hungary were ruled by the same king, but in this writing I considered them separate), towards mainly the Southern territories.