can you determine states/ countries people moved to when they migrated from a particular county (50) and by age

using 2010-2014 ACS data

There is not a variable that identifies this directly, however it should be possible to do something similar to what you explain above. First note that it is not possible to identify migrants by county. The lowest level geography in modern public use files is the PUMA (Public Use Microdata Area). Next, although a combination of MIGPLAC1 and MIGPUMA1 will identify state-PUMA combinations of residence a year ago, this is not exactly the information you seem to be looking for. However, if you include the PUMA variable in your data extract you should be able to identify the current PUMAs of individuals who resided in a particular state-PUMA combination a year ago.

I hope this is helpful.