Hypertension is one of the greatest contributors to premature morbidity and mortality worldwide. It has been demonstrated that lowering blood pressure (BP) by just a few mmHg can bring substantial clinical benefits, reducing the risk of stroke and ischemic heart disease. Properly managing high BP is one of the most pressing global health issues, but accurate methods to continuously monitoring BP at home are still under discussion. Indeed, the BP for any given individual can fluctuate significantly during intervals as short as a few minutes. In clinical settings, the guidelines suggest to wait for 5 or 10 minutes in seated rest before taking the measure, in order to alleviate the effect of the stress induced by the clinical environment. Alternatively, BP measured in the home environment is thought to provide a more accurate measure free of the stress of a clinical environment, but there is currently a lack of extensive studies on the trajectory of serial BP measurements over minutes in the home setting. In this paper, we aim at filling this gap by analyzing a large dataset of more than 16 million BP measurements taken at home with commercial BP monitoring devices. In particular, we propose new techniques to analyze this dataset, taking into account the limitations due to the uncontrolled data collection, and we study the characteristics of the BP trajectory for consecutive measures over several minutes. We show that the BP values significantly decrease after 10 minutes minutes from the initial measurement (4.1 and 6.6 mmHg for the diastolic and systolic BP, respectively), and continue to decrease for about 25 minutes. We also describe statistically the clinical relevance of this change, observing more than 50% misclassifications for measurements in the hypertension region. We then propose a model to study the inter-subject variability, showing significant variations in the expected decrease in systolic BP. These results may provide the initial evidence for future large clinical studies using participant-monitored BP.