Dubai is quit a distance from the UK. Any box that is to be shipped between the two countries will have to go by air. There is the British isle to be crossed. But after the British isle, every other route will have to be by land. Instead of having to switch between water way and land transportation, many of the shipping agencies only provide air transport to Dubai. The price you have to pay for the shipping is totally dependent on the shipping agency. But there is a range within which the shipping is done.
Some shipping agencies may be desperate for customers. Because of this, they will make their services available at a cheaper rate. If you search the internet very well, you should not have any problem in coming by one or two shipping agencies that may be willing to give you their services at a very cheap rate; cheaper than what you can possibly get from several other shipping agencies.
Be very careful how you package the box you want to transport box to Dubai from the UK. The packaging goes a long way to determine how much you end up paying for the shipping. If things are well packaged, you may be able to knock off a few pounds on the amount you will pay for the shipping. Instead of packing several boxes, you can decide to stuff as many of the things you have to transport to Dubai as possible in a single box.
No matter what you do, you may still end up paying hugely for the packaging. Many of the box shipping service providers charge you based on the weight of the box you are transporting. Packing them into one box or using more than a box may end up not making much difference.