Q:

How does one find and copy files of the same extension, in different directories, to a single directory in linux? 

So,  How Do I find and copy all files, 
*.a 

that are in, 
 ~/DIR{1,2,3,...} 

to 
 ~/tmp/foo? 

A:

Assumed you meant recursively copy everything of type .a from some source location. 
Haven't verified yet, but this should do that. 
find <root-of-search> -type f -name '*.a' -exec cp {} /tmp/foo \;

replace  with the top of wherever you want to search from.  You might have to throw quotes around *.a, and you might have to replace escape the ending semicolon by putting it in single quotes rather than back-slashing it.