Have a read of this link. Don't fuss too much over the equations - rather read the text and think about what it might mean.
pressbooks-dev.oer.hawaii.edu/atmo/chapt...ange%20%28neutral%29.
After you have done that, perhaps you might come back with some more specific questions.
Basically, while there is a tie up between convection (or convective flow) and adiabatic expansion/compression processes, they are two aspects of the things which happen in fluids where temperatures and densities are varying.
The term "convection" is tied up with the bulk vertical motion of fluids when there are dissimilar densities involved (think "buoyancy").
The term "adiabatic" just means that there is no energy transfer to/from the air parcels going on while parcels of fluid are expanding or compressing. As work still has to be done to expand or compress a fluid, there must be some energy used to make that happen. If we can't get it from the surroundings, it has to come from the parcel of air - that energy change then is tied up with temperature changes - in an adiabatic expansion, the temperature of the parcel of air drops while, for compression, it increases. Think of the initial compression of air in a pushbike pump if you put your thumb over the end hole - the air gets quite hot (associated with adiabatic compression). When you let the pressure escape, the temperature drops significantly (associated with adiabatic expansion).
Adiabatic processes are not specifically related to dew point. That is just where the air becomes saturated and we observe condensation, ie cloud base, and the temperature rates change due to the effect of what is called "latent" heat.
Edit - comment added for clarification