Enthalpy change vs. change in internal energy in a system
Both $U$ and $H$ are not directly related to any particular process.  They are physical properties of the material being processed, and are unique functions of the thermodynamic state of the material, and not the process.  They are directly proportional to the amount of material, and its thermodynamic state is characterized by its temperature and pressure (and composition for a multicomponent system).  The internal energy is a fundamental property, but the enthalpy is a defined property given by $H = U + PV$, and is just a convenient parameter to work with is solving many kinds of thermodynamics problems.

For a process carried out at constant pressure in a closed system, the amount of heat $Q$ added to the system turns out to be equal to the change in enthalpy between the initial state of the material and the final state of the material.  For a chemical reaction carried out at constant pressure, the amount of heat $Q$ that must be added to the system in order for the final temperature of the products to equal the initial temperature of the reactants turns out to be equal to the enthalpy of the pure products minus the enthalpy of the pure reactants at that temperature.  Both these results follow from the first law of thermdynamics.

For a chemical reaction, generally both the internal energy and the enthalpy change between the initial and final equilibrium states. 


  And what about cases where $P$ is not constant? What happens then? Can we calculate the value of $ΔH$? And does $ΔH$ even have a meaning in this case?


Since $H$ is just defined as $U + PV$, we can always calculate the change in enthalpy between two thermodynamic equilibrium states of a system.  $\Delta H$ certainly has a value in this case, even if it is not equal to the amount of heat added for a process between the two states.

The critical thing to remember is that $U$ and $H$ are physical properties of the materials comprising a system, and are independent of any specific process.