Switching between tunings mid performance

When I switch to drop D, it takes a few minutes of playing before the tuning stabilizes, and the same thing happens when I tune back up to standard. If I am performing, say, Capricho Arabe after a piece in standard, how do I avoid the D getting sharp halfway through? I find the inevitable going out of tune so bothersome that I have been avoiding any material in alternate tunings. Any and all advice is appreciated.