How can you tell, remotely, if someone put their SIM card in a new phone?
The SIM card has an international mobile subscriber identity (IMSI) that is used to tell who is calling.  SIM cards identify the owner.
The phone has an international mobile equipment identity(IMEI) that is used to identify the phone.
Both are used at various times when a phone is connected to a network.  That information will be stored for some length of time for billing and other purposes.
If you have the authority (court order and such) then the network operators can tell you which IMSI was used with which IMEI, and when (provided they still have the records.)
You could then match the IMEIs of the two people to the IMSI of the one person and see when each combination was used.
If all you have is the IMSI, then you can still get the records.  If the SIM is used for a long time with one particular IMEI, then changes to a different IMEI at about the time the murder (or whatever) occurred, then you have a hint that the killer may have swapped the SIM.