My dill plants leaves getting smaller as they are climbing upwards. I have been using a tomato cage for support.
That is a cucumber or related cucurbit vine. What you are seeing is normal for them. Dill is a relative of the carrot and parsley, and it will have a similar top growth and leaves. They will remain on the ground with lacy leaves and a basal rosette growth habit. See the articles below.
https://www.gardeningknowhow.com/edible/vegetables/cucumber/tips-for-growing-cucumbers.htm
https://www.gardeningknowhow.com/search?searchTerm=%22Cucumber%22
https://www.gardeningknowhow.com/edible/herbs/dill/potted-dill-plant-care.htm
https://www.gardeningknowhow.com/search?searchTerm=%22dill%22