Both retainers and braces can move teeth. The secret to moving teeth is very small amounts of pressure applied constantly. Both appliances can provide the pressure part of the process. The braces are more reliable on consistency of effort. Retainers, be they with wire or clear aligners are removed for eating, brushing etc. At best they are not as consistent as braces as applying the needed pressure.
The retainer can do the job, but it will usually take longer.
The braces can also be adjusted more accurately than a retainer. The spring of a retainer is bent and re-bent every time it is placed on the teeth. This can change the pressure level of the spring being used to place pressure on a tooth.
Even for small touch-up treatments, often braces for a few months can be far more effective than a treatment retainer used for 6 months. Also, braces can be used to make space for a blocked tooth far more easily than can a retainer. Lastly, braces can change the vertical position of a tooth. Retainers are very limited in this type of movement.