You don't say that this is new behavior, but I have to assume that this hasn't been going on for all these years (at least I hope not!) and that it is relatively new, especially since you use words like "familiar" and "comfortable."
When this situation crops up seemingly out of the blue, it is always important to look first for changes in her life. Start at home: has someone been sick, hospitalized, dying? Is someone stressed from work or from finances? Are you moving, changing jobs, fighting with your spouse? Is there a new baby? Is it possible your child is sick? Examine everything you can think of. Next look at the school setting. Is there a change in staff, friends or routines there? I'm sure you've already asked the teachers if anything is going on.
If you can't find anything -- and I'm betting you won't -- I would assume that this is not about anything past or present but future: she's anxious about kindergarten and sad about leaving the only setting she's known other than home.
May is prime time for a child's fears to surface, as preschool teachers (mistakenly, in my opinion) spend too much time talking about "getting ready for kindergarten" and saying things like, "You can't do that in kindergarten!" Even seemingly innocuous comments from the postman or grandma -- "You're so grown up! You're going to be in kindergarten!" -- can be ominous to a child who has no concrete idea what it means to be "grown up" or to "be in kindergarten."
So why would this behavior surface at drop-off? Because it's a daily reminder that she won't be coming here soon. She's old enough to understand that change is coming ("Mama won't bring me to this place next year!") but developmentally unclear about what it means. Her way to cope is to be upset that drop offs here will be ending. She may even say things like, "I'm not going to kindergarten!"
Here's what not to say: "Don't be silly, of course you're going to kindergarten." Don't say, "Don't you want to be a big girl, like everyone else?" Because right now, no, she doesn't.
Instead, validate her feelings and give her her wish in fantasy: "You really don't want to leave preschool, do you? You've been here since you were a baby!" "It would be nice if you could stay here always, wouldn't it?" Even: "I'm feeling sad to leave, too, we've been coming here all these years." If she doesn't verbalize it, put it out there for her: "I wonder if you're feeling sad to be leaving your preschool?"
Get out photos from years past, maybe even make an album together. Go around to the rooms she's been in and take pictures of them as they are now. Ditto for all the teachers she's known, toys she's loved, books she remembers best. Yes, this may feel like wallowing. It's also providing closure. In one family I know where the child had been in the same program since infancy and, like your daughter, was having drop-off problems, the youngster took a video camera to each classroom and (jerkily) focused it on "her" sleeping corner, her cubbies, etc. When she finished making the video, she announced to her parents, "Now I can go to kindergarten." Like magic, she stopped having issues. I"m not saying this will be a cure, but I suspect that at some point, on her own, she will start to move forward. You can nudge her along by driving past the elementary school: "Look! After summer time, that will be our school." Later, go to look and then play at the playground. But expect there will be blips in the road all summer long as the transition to kindergarten unfolds.