There shouldnt be any issue, dont forget to change the slug in the template part call:
get_template_part( 'format', get_post_format() );
the best practice is to have them in their own folder:
get_template_part( 'template-parts/post/format', get_post_format() );
you can see the folder structure of twentyseventeen just change content respectively.
How get_template will work:
get_template_part will do a PHP require() for the first file that finds, it will create an array the first item array will use slug and name like this:
{$slug}-{$name}.php
the second item array will be:
{$slug}.php
the function will iterate the array of file names till it finds the one that exists.
so if you do a:
get_template_part( 'format', get_post_format() );
and for example the format is video the array of file names will be:
[0] "format-video.php"
[1] "format.php"
it will fallback to format.php, in the case of a Child theme the priority will be like this, with the video format as example:
[0] "format-video.php" //in child theme
[1] "format-video.php" //in parent theme
[2] "format.php" //in child theme
[3] "format.php" // in parent theme