On the run

On the run
 

With all these IAish thingamagigs loose on the Internet, I'm starting to distrust much of what search engines and dictionaries offer me.

Please, confirm if I'm right

sb is on the run = alguien se ha escapado/ está huyendo. ("está huído" diríamos acá - huído con tilde, porque la Argentina es país del hiato)

keep sb on the run = tener a alguien de aquí para allá (haciendo cosas)/ en constante actividad.

You've used and interpreted both turns of phrase (the same turn of phrase, used in distinct ways) correctly.

Thank you, Rusty!