I found all the guys who had 150+ IP in the years 2003-14. If a guy pitched for two teams in a year, that guy in that year was disqualified. I found the year to year correlations for SO, BBs, HRs and BABIP (SO, BBs, HRs were all as a percentage of batters faced, which excluded IBBs and I included HBP in BBs). I used the Baseball Reference Play Index. (June 11-I made some corrections as mentioned in my first comment)
There were 488 cases of a guy being on the same team the next year (of course, if John Smith was on the Reds from 2011-13, we have one case of his 2011-12 seasons and another case of his 2012-13 seasons)
There were 107 cases of guys being on a different team the next year. Here are the correlations for each group
BABIP 0.094 (p value is .33)
The p-value in all cases except for the one listed are under .01. So the only one which is not significant (and it is not even close) is BABIP for guys that changed teams.
Now the correlations are lower for the guys that switched teams for all four stats. But the difference for BABIP is by far the biggest.
It is not clear what causes this. The pitchers have a whole new set of fielders behind them and they pitched half their games (or around that) in a completely different park than the year before. It seems like if the pitchers had the most control over what happens on balls in play, the correlation for guys that change teams would be closer to that of guys who stayed on the same team.
Phil Birnbaum suggested I run a regression in which BABIP in year 2 depends on BABIP in year 1. Here it is for guys that stayed on the same team.
BABIP2 = .234*BABIP1 + .226
No the regression for guys that were on a different team
BABIP2 = .093*BABIP + .268
The effect is 2.53 times higher if you stay on the same team. That is, if you stayed on the same team, your BABIP in year 1 is about 2.5 times stronger in predicting your BABIP in year 2 than if you switched teams.