How should one proceed with an exchange?

General Procedure for Replacing Capacitors

First, it is advisable to take plenty of photos: of the wiring, the circuit board, and the capacitors.

Electrolytic capacitors must be installed with the correct polarity. In your case, the negative terminal is usually marked by a stripe along the side of the housing.

It is helpful to be able to check how the capacitor you are holding between your fingers was installed, so that you can install the new one the right way around (in some cases, you also cannot rely on the markings printed on the circuit board, if present).

I recommend desoldering with the help of desoldering braid and a desoldering pump. An electronic desoldering station with a heated suction gun would be best, but in my opinion, purchasing one is not worthwhile for smaller projects.

First, assess the situation. Keep in mind that in the vast majority of cases, electrolytic capacitors with a higher voltage rating will be supplied.
The maximum permissible voltage rating (V) may match that of the old capacitor, but it can and ideally should be higher and must never be lower

An example:

Your original device contains four different types of 47µF capacitors:

47µF/10V,

47µF/25V,

47µF/35V, and

47µF/50V.

 
As replacement types, we would supply you with the following:

2x 47µF/35V as replacements for 10V and 25V

2x 47µF/63V as replacements for 35V and 50V

Although the original includes a 35V capacitor, a 63V capacitor is supplied instead so that the voltage rating is higher.