Total differentiation

I would rewrite your equation in a more natural order, at least to me:

$$frac{DT}{Dt} = frac{partial T}{partial t} + mathbf{U} cdot nabla T $$

The term $frac{DT}{Dt} $ is the total derivative of $T$ with respect to time, i.e. how temperature changes (in one particular place on Earth) with time passing, which can have different reasons. The term $mathbf{U} cdot nabla T$, as you point out, is the advective term. It expresses that winds can bring warmer or cooler air onto this place, making it cooler or warmer there, such as cold fronts.

The term $frac{partial T}{partial t} $ expresses the $local$ change of temperature with time. If no winds are blowing, temperature may change as well, for example due to insolation (cooler at night, warmer during the day), seasonality and so on.

Therefore, the term $frac{DT}{Dt}$ on the left-hand side of the equation makes it maybe it more clear. It is not a component of the derivate, it is the total derivative, decomposed in local and advective terms. So advection is one of the components of $frac{DT}{Dt}$.

As far as the question "What can be causing this?," there are a variety of reasons. Let's look at the First law of Thermodynamics (including molecular diffusion): $$frac{DU}{Dt}=c_pfrac{DT}{Dt}=Q-pfrac{Dalpha}{Dt}+nu nabla^2 U$$, where $U$ is the internal energy of the ideal gas, $Q$ is diabatic heating, $p$ is pressure, and $alpha$ is the specific volume, $nu$ is the molecular diffusivity, and $c_p$ is the isobaric specific heat capacity.

So, without diabatic heating, one way is to change the volume or pressure, which varies the most with changes in altitude (you can rewrite the $-pfrac{Dalpha}{Dt}$ term as $alpha frac{Dp}{Dt}approx alpha frac{partial p}{partial z}approxalpharho g=g=-Gamma_d c_p$, $Gamma_d$ is the dry adiabatic lapse rate). Using that relationship, (1) may be rewritten as $$frac{partial T}{partial t}=frac{Q}{c_p}+nunabla^2T-frac{g}{c_p}-Gamma_d- vec{u} cdotnabla T$$ Another way is through molecular heat exchange, which is so small it is usually neglected, unless it is in a micrometeorological context. And then there is diabatic heating, perhaps the most loaded term of the equation.

Diabatic heating is any heating source that requires entropy to change. This includes latent heat exchange, radiative heating, etc. Turbulent mixing may arguably be considered in this for large scales, but not small scales.