I think one common way is to get a high ranking bureaucrat to mandate the change. It is not a guarantee, but a good enabler. I suspect pride, shame (you still use paper receipts for this?), media or even bribes are used.

For examples, I would look at automating standard government functions (e.g., car registrations, toll collections, etc.). Putting those online leaves some clerks unemployed and is thus often resisted by default, so someone found a working path for each of those

I think competition among countries actually helps. Case in point: this visa program. They are trying to make it more streamlined and more attractive with a longer expiration. It is improvement in the right direction.