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In case of a constant current source at first the current will flow at contant rate, so the voltage increase will be linear, but if it is not ideal case where the supply voltage is infinite, finally, the current will decrease --> it asymptotically approaches zero for longer, and longer times. Similarly the charge (voltage) will increase and asymptotically approaches (but never actually reaches) its maximum value. It would take an infinite amount of time to actually reach the maximum value (ie. supply voltage of the non-ideal current source) since the rate of increase (current) becomes smaller and smaller as time goes by.
In case of a voltage source, connecting a capacitor to it in series and not closing circuit will have no effect at all (DC voltage) ..
If you close the circuit through "someting", depending on the circuit, it may behave as normal RC circuit ..
See capacitor charge at: https://hyperphysics.phy-astr.gsu.edu/hbase/electric/capchg.html ..
What happens if you connect a capacitor in parallel with a current source depends on the rest of the circuit. The AC impedance of a current source is large (depending on the configuration) but the capacitor will pass the AC signals forming a high pass filter with the internal resistance (parasitic) of the current source. From a DC point of view if the rest of the circuit does not draw current the capacitor will charge linearily to the saturation voltage of the source.
When you connect a capacitor in series with a voltage source you block the DC current. However the AC current will pass and since the AC impedance of the voltage source is low AC current will flow to ground. What kind of configuration do you have in mind?