There were many reasons why there was a major decline in childhood deaths - improvements in clean water and sanitation, changes in public hygiene, general socioeconomic improvement, improved nutritional status all were factors in the mortality decline. These improvements and resultant decline in childhood deaths occurred well before other interventions such as immunizations.
This is precisely why you have a tremendous amount of children dying from measles in developing countries.
From the World Health Organization (WHO):
Why children die of measles in Africa
Measles, itself, does not kill children. Instead, complications from measles attack their already weak immune systems. Measles attacks the body, inside and out. It is similar to HIV in the sense that when it knocks down the immune system, the child becomes susceptible to the myriad of diseases festering in poor living conditions. While the typical red spots of a measles rash in America may signal a mild disease, that same rash becomes a severe attack on all of an African child's skin surfaces including the gut, cornea and lungs. The direct damage caused by measles can be high fever, peeling of the skin, and encephalitis leading to brain damage. Complications from measles are even more severe and can include blindness, severe diarrhea, and pneumonia. Measles is just one more assault on their already unhealthy bodies.