As a team lead I insist on a link to a ticket for stuff that is related to a ticket. When moving to another ticket system, I insist either on keeping older projects on the old system, or alternatively on migrating old tickets to the new system and keeping the original references; the same way I insist on keeping history when moving from one source control system to another one.

Also... why would a company go through 3 different ticket systems in 5 years? That must cost a lot and definitely is not gonna help convincing people to use it...