Yes, swelling is common. Expect it soon after the injection and watch whether it eases over the following days to weeks.
Two published sources describe the same pattern, and both are plain about it.
What the FDA and NCBI PMC say
The FDA states that most side effects associated with dermal fillers, such as swelling and bruising, occur shortly after injection and many resolve in a few days to weeks. NCBI PMC says swelling is a common complication after filler injection, as hyaluronic acid attracts and binds water to the tissues.
Put together, that means swelling is an expected early effect and the stated reason is water binding. The word "many" is doing work in the FDA sentence. It does not say all, so swelling that is not easing is a question for your provider.
Other pages on fillers and lips
For general background, read OHSU's page on dermal fillers and UCLA Health's page on non-invasive facial rejuvenation. Lip work is covered in Cleveland Clinic's guide to lip augmentation, fat transfer and fillers, and the ASPS article on med spa treatments is a further reference.
Knowing who to call about swelling
Ask your provider before the visit what swelling they consider expected and who to call if yours looks different. Put that number in your phone before you leave the office.