I believe this goes both ways

Just this side of heaven is a place called Rainbow Bridge.

When an animal dies that has been especially close to someone here, that pet
goes to Rainbow Bridge.
There are meadows and hills for all of our special friends so they can run
and play together.
There is plenty of food, water and sunshine, and our friends are warm and
comfortable.
   All the animals who had been ill and old are restored to health and
vigor; those who were hurt or maimed are made whole and strong again, just
as we remember them in our dreams of days and times gone by.
   The animals are happy and content, except for one small thing; they each
miss someone very special to them, who had to be left behind.

They all run and play together, but the day comes when one suddenly stops
and looks into the distance. His bright eyes are intent; His eager body
quivers. Suddenly he begins to run from the group, flying over the green
grass, his legs carrying him faster and faster.

You have been spotted, and when you and your special friend finally meet,
you cling together in joyous reunion, never to be parted again. The happy
kisses rain upon your face; your hands again caress the beloved head, and
you look once more into the trusting eyes of your pet, so long gone from
your life but never absent from your heart.

Then you cross Rainbow Bridge together....
*** " And nothing but peace and the love of your Dear Friends"****

Author Unknown

This should be added   ***
****"Always remember We will be there even if you can't see me. We are
always watching and Waiting.  That cool puff of air you feel, that fleeting
touch , the feeling you have that I was walking across your bed, that moment
you swear you can hear us, that flicker of movement out of the corner of
your eye.  Is just our way of saying I love you and I am with you always
even in the darkest time We Will Always Be there"***

Author
Matthew ( NO MORE RETAIL )

In that case you pass over normally, but your pet is directed to the
pearly gates (human equivalent of rainbow bridge), where you 'know' to
meet him/her.

It's a nice fairy tale, isn't it?

Were I to be the forst one to go, I will keep an eye on her for the
rest of her natural life and be right there at the Rainbow Bridge when
she comes.

It is not necessary to assume that time (sequence) has the same
meaning it does in our world - if it has any meaning at all.