I understand most airlines allow 2 pieces of luggage at up to 23 kg each. However what happens if I have got more than that? I am travelling to Zimbabwe using either Kenya airways or Emirates.
What will happen depends on the airline, your travel class, your frequent flyer status, the time of the day, the mood of the check-in agent and a lot of other things.
In principle, from the moment you have more than your luggage allowance, either because you want to give up an additional piece (typically the third) or because your pieces are heavier than allowed, you can be subject to additional charges or the airline can refuse to accomodate your luggage. However, it is common for airlines (not low-cost carriers) to waive additional charges and still take your luggage if it is overweight. The higher your class and status are, the more likely. So flying business class, accompanying a gold member, Lufthansa would allow me two suitcases at something like 28 and 25 kg. If I had taken economy, they may have decided to charge me extra because of the obviously too heavy bags.
I hear (but cannot confirm from my own experience because I never flew with them) that Emirates gives a rather large luggage allowance in the first place; something closer to 30 kg. However, they will allow only one bag and be very strict when it comes to overweight — to the point that it may be cheaper to buy a second ticket rather than an additional bag. I cannot give any information about Kenya Airways.
In short, you should check with the airlines individually. Usually, the terms and conditions will be posted somewhere on their websites. You may also opt to call them. Unfortunately though, they will likely not tell you how lenient they are.