Peel the rubber back a little and use the straw to blast WD 40 in any opening you can.
Better still, turn the bike upside down and use the straw to shoot it into the works.
Use lots, clean the levers to remove the excess WD40 from the outside.

Shifting problems caused by worn or dirty shift cables and cable housings can act similar to gummed up shifters.
The cables and housings should be replaced occasionally. The short housing at the rear derailleur is most critical.

Recently I had shifting problems in the front due to dirt and grime in and around the pivots. This could also be the cause of sluggish shifting at the rear derailleur. I soaked the FD in mineral spirits and sprayed WD40 on the pivots. Problem solved.