Switching narratives has become a common writing feature and used well, it can be very effective.
Switching perspectives allows you to tell a story through the two separate narratives, which usually come together towards the end.
Grammatically you don't have to change anything between narratives, but you could alter person, tense and place on a timeline.
For example, in the story I'm currently working on, there's a main narrative in first person limited (I.e. 'I'), past tense with the protagonist, set as 'now'. And a second narrative in third person omniscient, past tense, set as 'then'. (See the chapter on 1st , 2nd , 3rd person; limited + omniscient if you're confused)
So, you can change from first to third person (I.e. 'I' to 'He'/'She').
Or tenses (I.e. from present 'are' to past 'was' or future 'will') as you switch between narratives.
But, you don't need to switch tenses to move into the past/future.
The contrasting narratives could be told by the protagonist and antagonist (See chapter character types: protagonist and antagonist if you're confused).
Switching perspectives can be confusing for a reader, but used well creates hooks. (See chapter hooks if you're confused)
Differentiate your 'main' narrative from your second one/others with distinct chapter headings and contrasting chapter lengths.
Don't have too many different narratives at once. This could become confusing. Generally I'd say two is enough with a possible third for exceptions. Unless your plot is heavily planned and requires several perspectives, I'd try to avoid over-use of it.
Do you use this writing device in your work?