How do I migrate while content is still created on production site?
Migrate, as shipped with Drupal 8, has two components:


The migrate module. This provides a very flexible API. 
The migrate_drupal module. This provides Drupal 6 / 7 to 8 migrations (the migrations themselves has been scattered into the modules). This is the part which keeps IDs and the motivation for this was very simple: there's just no way to tell where those IDs might lurk. Say, one of the old nodes has a http://example.com/node/12345 link in the body text and we changed the node id to 67890 during migration? Then that link breaks. It also made for more rapid development which is not a bad side effect either. It is arguable whether for example file IDs deserved such treatment or not.


As @berdir comments, if you remove the nid: nid process definition from a node migration then it will happily generate new node IDs for you, no sweat but you need to look everywhere for the same to make sure that any references are changed to a migration_lookup process definition. Some of it is already done for you, some of it is not.

For example, comment/migration_templates/d6_comment.yml has:

# If you are using this file to build a custom migration consider removing
# the cid field to allow incremental migrations.
cid: cid
pid:
  plugin: migration_lookup
  migration: d6_comment
  source: pid


For core, pid: pid would work just as well but instead, this migration is already prepared for this migration creating new IDs. But then it continues with

entity_id: nid


Now the corresponding node/migration_templates/d6_node.yml has

# If you are using this file to build a custom migration consider removing
# the nid and vid fields to allow incremental migrations.
nid: tnid
vid: vid


If you actually do that then you need to circle back to comment/migration_templates/d6_comment.yml  and write

entity_id:
  plugin: migration_lookup
  migration: 
    - d6_node:article
    - d6_node:page
  source: nid


Because the actual migration is not d6_node but rather d6_node:nodetype.