[Bridget] kept getting ahead of me (not pulling, but very much focused on what was in front of us), and I felt (perhaps mistakenly) that she wasn't maintaining an awareness of my presence. So every so often I asked her to do something to remind her that I was on the other end of the leadrope.
Jenny aptly commented:
[H]orses are aware of all things going on around them, people (with their 'tunnel vision', comparing to horses!) tend to focus all their attention in only one thing at a time. So when our attention is pointed away, horses tend to feel they have to take care of themselves as we are busy doing something else.
Today I took Bridget out for a little walk by herself (Chloe having declined to accompany us). This time I was determined not to "mind" if she got ahead. On our last walk, I was busy focusing on Bridget, while Bridget herself, of course, is always very attentive to everything going on around us. This time I tried to not just pay attention to her, but to notice what she was noticing. I might not see exactly what she sees, but at least I can look in the same direction.
I observed that she doesn't just barge ahead willy-nilly - that she will make a slight adjustment in her stride to wait up for me a little, and every now and then she will turn and "consult" by gently touching me with her nose.
On our way back down the drive to the barn, I proved what a numbskull I am. Bridget suddenly turned around - there was a horse and rider coming up behind us which I totally hadn't heard. Guess Bridget is the right one to be leading after all!