I've been going back and forth between standard, open E, and D tuning on some songs I play. A lot of the time strings will break even when I'm tuning down. Is this because of the truss rod? Are there any adjustments I can make to save my strings as I go from one tuning to another?
Your truss rod does not touch the strings or affect the tension on them, so it cannot be responsible for breaking your strings. If there is some problem with your guitar that is breaking the strings, it will be found where the guitar touches the string. If the strings tend to break at the nut, it's probably the nut breaking your strings. The same idea applies if they break at the bridge, ball end, tuning machine, or elsewhere. Once you locate the problematic part you can have that part repaired or replaced.
I like Edward's answer and totally agree, but I want to add details.
String Quality and Metal Fatigue — I've had strings break in first tighten, not overtightened. Bad strings pass inspection. It happens. And with retuning and such, strings can break. Were the new strings? From a good brand?
The Nut and the Saddle — These are the two likely places where there's breakage, because these are where the strings touch the body. You retune, which implies a hardtail, but is it acoustic or electric? And does the same string break, or different ones? Where does it break; the nut or the saddle? I would guess that it's electric with a metal bridge, always the same string and near the bridge. There is likely sharp points where the string rubs and breaks. It is that commonly a problem. Nuts are commonly bone or plastic, which should give before metal strings, but I'm sure it happens. If we get more details, we can give better answers.
I now see the acoustic-guitar tag. That negates much of the last paragraph and makes me think old strings giving up is the culprit.
The truss rod exists entirely to keep the neck straight when confronted with hundreds of pounds of tension from the strings. It does nothing that could break strings. It is not the problem.