Python 2, 53 bytes

lambda a:reduce(lambda b,v:b+[b[-1]/v*v+v],a,[0])[1:] 

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Python 2, 53 bytes

lambda a:reduce(lambda b,v:b+[b[-1]/v*v+v],a,[0])[1:] 

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k*x>y implies k>y/x; so the smallest k can be is k=floor(y/x)+1. Since in Python 2.7, integer division is already taken as floor, we want k=y/x+1, and k*x = (y/x+1)*x = y/x*x+x.

Python 2, 55 bytes

lambda a:reduce(lambda b,v:b+[v*(b[-1]/v+1)],a,[0])[1:] 

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Python 2, 53 bytes

lambda a:reduce(lambda b,v:b+[b[-1]/v*v+v],a,[0])[1:] 

Try it online!