The MALR (Moist Adiabatic Lapse Rate) is also called the wet or saturated adiabatic lapse rate. It is the temperature trajectory a parcel of saturated air takes. The dry adiabatic lapse rate is a near constant of 9.8 C/km, however, the wet adiabatic lapse rate is much less of a constant. The wet adiabatic lapse rate varies from about 4 C/km to nearly 9.8 C/km. The slope of the wet adiabats depend on the moisture content of the air. The more moisture (water vapor) that is in the air, the more latent heat that can be released when condensation takes place (the release of latent heat warms the parcel while an absorption of latent heat cools the parcel). Any warming by latent heat release partially offsets the cooling of rising air. Notice on the skew-T that the dry and wet adiabats become nearly parallel in the upper troposphere. This is due to the very cold temperatures aloft (cold air does not have much water vapor and therefore can not release much latent heat) The slope of the wet adiabats is 4 to 5 C/km in very warm and humid air (lifting of this saturated air releases a large amount of latent heat). Warm and humid air in the PBL contributes to atmospheric instability. These warm and humid parcels, since they only cool slowly with height, have a good chance of remaining warmer than the surrounding environmental air and will thus continue to rise. In fact, planetary boundary layer warm air advection and moisture advection are the number 1 contributions to making the troposphere thermodynamically unstable (High CAPE, negative LI, etc.).
The formula for the moist adiabatic lapse rate is
MALR = dT/dz = DALR / (1 + L/Cp*dWs/dT)
Every term in the equation is a constant except for dWs/dT. dWs/dT is the change in saturation mixing ratio with a change in temperature. The saturation mixing ratio changes at the greatest rate at warm temperatures. Increasing the temperature from 80 to 90 F will change the saturation mixing ratio more dramatically than changing the temperature from 30 to 40 F. Thus dWs/dT is higher in warm air. As dWs/dT becomes larger, the denominator in the MALR equation becomes larger and thus the MALR becomes less. Math example: 1/4 is a smaller number than 1/3 because the 4 in the denominator is larger than 3. In very warm and moist air, the MALR will be near 4 or 5 degrees Celsius per kilometer. At very cold temperature, dWs/dT is small, thus the denominator is close to one and the MALR is close to the DALR (9.8 C/km). When dWs/dT approaches zero, the denominator becomes 1 and the MALR = DALR.
The formula for the saturation mixing ratio is: Ws = 0.622Es / (P - Es). Therefore Ws depends on the pressure and Es of the air. It is temperature that determines the moisture carrying capacity of the air. Remember that Es is found by plugging T into the Clausius-Clapeyron equation. Therefore, ultimately, Ws depends on temperature and pressure.
If instability is present, the instability will increase further when the PBL experiences rising dewpoints (above 55 F and rising). Thunderstorms are much more common in the warm season. Warm and moist rising parcels of air do not cool off as fast as rising parcels of colder air. Since warm and moist rising parcels cool at a slower rate with height (due to more latent heat release than colder air), the parcels are more likely to remain warmer than the environmental air and rise due to positive buoyancy.