The MTU will change VERY infrequently on the interface of a production router. In fact it is likely that the MTU would NEVER change on the interface of a production router.
just to add to Rick's post:
usually MTU is changed in a production network only after careful planning.
I think the primary reason for changing MTU in a network nowdays is MPLS introduction and in this case we talk of different settings (mtu, ip mtu, mpls mtu).
However, once you decide what MPLS services are going to run on the network if there is use of MPLS TE FRR, appropriate values for all parameters can be set and left there for long time.
Hope to help
It may as your cross different media. However, once on a particular media (physical or logical), or for the same kind of media, it's often same. I.e. its tends to be very static.
For instance, Ethernet and many WAN media support different MTUs. So if you go from LAN to WAN to LAN, the supported MTU might change from LAN to WAN to LAN. If the LAN on both ends is Ethernet, there's a good chance the MTU would be the same for them.
If MTU decreases along the path, and if a IP packet is larger, it can not be forwarded as is. It might be fragmented to fit the smaller MTU. If it is, if the MTU is later increased along the path, fragments are not recombined (until received by the destination).