Yes they will do just fine on starter. The only difference, other than the protein level, is the grower/finisher in my feed, is not ground as fine. I am feeding mash though. If you are on pellets, there won't be any difference to worry about.
As I understand it, the higher protein level in starter, is to help the bird to grow quickly. They need all of that protein as fast as they change in the first six weeks or so. The grower/finisher is used as a holding feed, to let the bird fill out to size without putting on excess weight. Of course I am speaking of laying birds not meat birds.
Somewhere between about 16 to 18 weeks you move to laying feed, and the formula changes once again.
Hope this helps some, the above is just best practice, not a hard and fast rule. Chickens don't care much, as long as they do get fed.
Best to you and your birds,
RJ