It is not possible to quickly export or import canned responses. Given that a project may have up to 200 responses, it would be relevant to be able to backup the responses and quickly export/import them into other projects/sites.

Workaround:
There is not currently a REST API endpoint for retrieving the canned responses, but it is possible using the Atlassian GraphQL gateway.

Here is an example GraphQL query to pull this data. Keep in mind that you will need to use the cursor that is returned for pagination to get all of the responses:

Query:

query GetProjectCannedResponses($projectAri: ID!) {
  jiraCannedResponse {
    searchCannedResponses(
      filter: {projectAri: $projectAri, query: ""}
    ) {
      edges {
        node {
          content
          title
          createdBy
          id
          isSignature
          lastUpdatedAt
          projectAri
          scope
        }
      }
      pageInfo {
        endCursor
        startCursor
        hasPreviousPage
        hasNextPage
      }
    }
  }
}

Variables:

{
  "projectAri": "ari:cloud:jira:xxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxxx:project/10001"
}

Note that in the projectAri variable xxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxxx is your Jira cloud site id and 10001 is the id of your project.

You can access the Atlassian GraphQL gateway via https://developer.atlassian.com/gateway/api/graphql or https://your_domain.atlassian.net/gateway/api/graphql

Documentation for the GraphQL gateway can be found here:
https://developer.atlassian.com/platform/atlassian-graphql-api/graphql/#overview