What is happening in this situation?
Since you did not provide a circuit schematic diagram or component values we have to guess what you are measuring. However you are describing the classic observations of the capacitor charging. 

"Resistance" readings from a simple DMM performed on a capacitor really has no useful meaning.  There are too many variables to even know what the readings mean.

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The voltage across the capacitor Asymptotically approaches the charging voltage. Here is an example of the classic capacitor charging curve: