Freezing Point Depression and Temperature Decrease
Phase change is not at constant temperature, phase equilibrium is


  [OP] it's a phase transition so the temperature of the ice should remain constant


Here is a counter example: If you add ice cubes to hot water, the ice will melt, cooling down the hot water. In this system, there is no thermal equilibrium, so not all of the components are at the melting point of water. If you wait a while, you might see the following: All the ice is gone, or some ice and some water remains, or (if the ice was really cold) all the water is frozen. Only if there is some ice and some water left do you know the equilibrium temperature - equal to the melting point.

A system can gain thermal energy from other types of energy

If you stir a viscous solution, it will get measurably warmer. (This is a classic experiment of thermodynamics.) There is no heat transfer - nothing gets colder. For melting ice, it is a conversion of thermal energy into chemical energy (breaking hydrogen bonds).


  How can the ice be cooling as it melts?


It can't, at least not for pure water (see below for adding water). The temperature of the ice is below the melting point (or at it), and the temperature of the water is above the melting point (or at it, or a bit lower if it is super-cooled). The thermal energy will come from the liquid water, cooling it. If everything is at the melting point, melting and freezing will happen at the same rate, and nothing changes temperature. In a real system where surrounding is warmer, thermal energy from the surrounding flows into the system, melting the ice while the system stays at the melting point (there will be small fluctuations so that the thermal energy reaches the ice).

Scenario of adding salt

Starting with an ice/water equilibrium (at melting temperature), we can add salt (at the same temperature, and let's say there is no enthalpy of dissolution). After adding the salt, the temperature is still the same, but the melting temperature is lower. With the temperature higher than the melting temperature, water molecules break away from the ice, which requires energy. This energy comes from thermal energy in the liquid and in the remaining ice. As a result, the entire system cools down. 

Will the temperature change?


  In your example where ice cubes were added to hot water, wouldn't the pure water ice cube remain at 32 F until it fully melted? How is this any different than having an ice cube on a hot plate and melting it? The temperature during the heating of the cube doesn't increase because the absorbed heat is being used to break the chemical bonds of the solid ice.


If the system is not at thermal equilibrium (different temperatures within the system), some parts of the system have to change temperature before equilibrium is reached. The hot water will certainly decrease in temperature. If the ice cubes are colder than the melting point at the start, they will warm up until they have reached the melting point. (Ice can have any temperature below the melting point. For example, ice from a freezer is at about $\pu{-20^\circ C}$.)

If the system is at thermal equilibrium from the start but not at phase equilibrium (or not a chemical equilibrium for a reaction that is endothermic or exothermic), the temperature will also change. In my example of adding salt to a water: ice mixture, the process of melting is endothermic, so the temperature at equilibrium will be colder (and equal to the freezing point of the salt solution). As ice melts, the temperature decreases and the freezing point increases (because the pure melt-water dilutes the salt solution). Once the two temperatures match, the system is at equilibrium (thermal equilibrium and phase equilibrium).