Your question about swelling and drooping of the eyelids 24 hours after a Dysport treatment leaves some relevant details out, so I will answer in several different ways hoping to address any possible issue. First, the results you are experiencing are not necessarily due to error on the part of the injector. However, without knowing how much volume you had, and exactly where the injections were placed, it's hard to be sure what the situation is. Though Dysport and Botox do start to kick in as quickly as one day for some people (but both increase effect over time), there is a good possibility that some of the swelling and fullness you are experiencing could be from the associated volume of sterile saline used to dilute and inject the Dysport as well. This is basically just water, and will be reabsorbed into your body over the next few days if it hasn't already. Weakening the "squeegee" orbicularis oculi muscle with lateral canthus treatments can make this fluid harder for your eyes to get rid of as quickly as normal.
However, if there really is a problem here and too much Dysport was injected too low in the forehead area, or just in the wrong spot, that could cause either a heavy brow feeling with fullness and the sensation that it is difficult to fully open your eyes, or possibly direct true lid ptosis (weakness) in which the brow is fine but the eyelid won't open. Either of these situations may worsen in the next 3 to 4 days as the Dysport takes full effect, and is difficult to fully reverse, but may be improved with prescription eye drops you can get from your doctor.