I found the following article.  This doesn't answer the question of whether a player becomes a starter, but it does give information about how many players drafted in each round are on the team that drafted them, by year. 

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I decided to go back over a decade and looked at each and every draft pick and tracked whether they were with their team each year for up to five years. I took all this data and created a complete table where I was able to see what percentage of players were on the teams that drafted them at the start of their 1st, 2nd, 3rd, 4th and 5th years. By creating this table we are able to get an idea of how long we can expect a given player to stay with the team that drafted them on average.

To read the table the rows represent the round the players were drafted in while the columns show what percentage of players were on the team after that specific year. For example 96.8% of all 2nd round picks are on the team's active roster that drafted them at the start of year one but only 83.9% at the start of year three.

Round Year 1... Year 2...Year 3... Year 4... Year 5
1........ 99.7%.. 93.5% ..83.9% ...77.4%...71.0%
2........ 96.8%.. 96.1%.. 83.9%... 74.2%...41.9%
3........ 96.9%.. 75.1%.. 62.5%... 37.5%... 18.8%
4........ 91.4%.. 74.3%.. 54.3%... 34.3%... 17.2%
5........ 81.1%.. 56.8%.. 37.8%... 24.3%... 16.2%
6........ 70.2%.. 57.5%.. 35.3%... 20.9%... 10.6%
7.........58.3%.. 45.8%.. 31.3%... 21.7%... 16.7%

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