It could be the idea of death has rattled her. Or, at this age DCs may have trouble determining between fact and fiction and an off-the-cuff remark by someone or a tv or film sequence that DD doesn't thoroughly comprehend may have set off this anxiety.
Acknowledge her anxiety, try and rebuild trust. Re-train her to accept Mummy and/or Daddy will be as good as their word, help her learn that you will do all you can to be on time and back to collect her. While you are away, you will do nice normal safe dull things, then you will return.
Plus, she is safe where she is, and grandmother or teacher will take good care of her in your absence.
One tip I heard was, before leaving the house for school, open her palm, draw an X, kiss it, curl up her fingers. There's a kiss for her, any time of day she needs it. When you get to school, ask her if she still has it. A routine can work its way into her head, distract her, reassure her. Or try a simple sticker reward system, one for every time you drop her off and she doesn't sob.
Maddening and stressful as it is, don't let on to her if you think it's annoying or babyish. She feels this way, it's all right to be worried or upset, but Mummy and Daddy can listen and help her feel better, take it seriously but stay calm, confident, to give her courage.
So, at home, practise going out of sight, into the garden or upstairs, telling her where you'll be and approximately how long before you're back next to her. Build up gradually. Get DH to do the same. No big farewell scenes at school, peel her off, assure her you'll be along to collect her.
Finish on a positive note,
"Mummy/Daddy will see you after school, you can tell us all about the new/clever things you've done at school today".