If I have ten servers, each with 100BASE-TX ethernet network cards, and I place a 1000BASE-T network switch between them, all servers are communicating as fast as they can, will the switch be handling at its capacity, or is it only bound to the network capacity of the 100BASE-TX ethernet cards, resulting in the switch operating at 100BASE-TX speed?
The secondary limitation in how a network switch performs (primary being the link rate between the switch and the device) is the switch fabric speed (usually expressed in gigabits) and packet forwarding rate (in millions of packets/sec). If all devices are connected at 100mbps not many modern switches would still have a bottleneck (they usually anticipate something close to the total of all ports at highest speed.) To directly answer your question: the switch will not constrain the traffic or act in a derated manner, the switch fabric speed operates independently of the link rates of any of the attached devices.
Each switch port will operate at the speed of the established link on that port. The ports are independent of each other when it comes to negotiating the speed. Ports where devices have connected at 1 Gbps will operate at 1 Gbps, but ports where devices have connected at 100 Mbps will only operate at 100 Mbps.
Where you introduce a bottleneck (e.g. traffic coming in a port at 1 Gbps, or multiple 100 Mbps ports, destined to single port which only has a 100 Mbps connection), you will probably drop a lot of frames. This can't be helped, and it is the way switches work.