I will just expand on what Chester Miller had already said.
The molar specific heat at constant volume $C_v$ is basically the amount of heat required to raise the temperature of 1 mol of the gas by $1$ K while keeping the volume constant.
When we apply heat to an ideal gas, some of that heat is used to increase the internal energy $U$ of the gas and the rest is eventually used up to do the work to increase its volume. (Note that increasing U essentially means increasing the temperature since U is a function of T for any ideal gas.) Now if we were to keep the volume constant somehow, then the total applied heat would be used only to increase the internal energy or so to speak the temperature and nothing else.
So a different way to define $C_v$ is to say, it is the amount of heat required to raise the temperature of 1 mol of the gas by $1$ K, where the said heat is used to only to change the internal energy U or so to speak the temperature and nothing else.
Now let's consider a case where change in volume is allowed. Say we applied $dQ$ amount of heat to the a gas. Now then we know some of that heat is going to account for the change in internal energy, let's say that amount is $dQ_u$. And the rest of the heat will eventually account for the change in volume, and say that is $dQ_w$. So instead of writing $dQ = dU + dW$ we write, $$dQ = dQ_u +dQ_w$$
So we can see that the heat $dQ_u$ is only used to change the internal energy and not the volume and so $dQ_u = dU$ and we can say according to the definition of $C_v$ the following $$C_v = \frac{dQ_u}{mdT}\ \Rightarrow\ dQ_u = mC_vdT\ \Rightarrow\ dU=mC_vdT$$
So, you see $dU = mC_vdT$ whether or not there is any change in volume. The same argument goes for the adiabatic process you mentioned.
