Hormone changes can certainly lead to changes in period color (or length, or heaviness, and so on!). So, while it's certainly possible, darker period blood is not a birth control side effect that I've heard of before. It sounds like a great question to ask the healthcare professional who prescribed the birth control. In the meantime, you might want to check out this article about birth control and possible side effects.
It can be. When you start a new birth control pill, it may have different hormonal effects and as a result the uterine lining may not build up as thickly as it has in the past. In addition, the hormones of your new pill may withdraw differently, and as a result the lining may slough off more slowly. Dark blood is old blood and the slower the lining sloughs off, the longer it takes for that little trickle of blood to get out through the vagina. The longer it takes, the older and darker the flow will be as a result. Depending on your body’s response to the hormones in the pill, you may never get a thick enough uterine lining to slough quickly and get red blood. Whether your menstrual flow is red or brown, it still counts as your period.
This is not intended to be medical advice. Everybody is different so please make sure to consult your physician if you're having issues. Do not delay or refrain from seeking professional medical advice from your physician because of something you have read on this site.
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