GOD is eternal, (exists outside of time's constraints), and never changes.
We are part of a grand expression of GOD during which He Is obscuring His Names, (attributes, wisdom and powers), then revealing them again. GOD Himself is not changing. The 'timeline' we are experiencing exists as a part of that same expression, but is only one 'arrangement' of reality, (which the Bible calls a 'kosmos,' Greek: 'orderly arrangement,' from which we derive our English word 'cosmos').
Imagine your 'pets' as creatures that exist per se, not only bound inside a particular arrangement or timeline. They 'begin' and 'end' only based on whatever arrangement and set of 'rules' GOD sets. Anything which no longer serves the purpose of continuing to reveal the aspects of His nature which are not yet revealed in the current 'arrangement' seem to vanish from our view and our experience. However, this does not mean that they vanish per se. That is why GOD calls death 'sleep.'
Whether GOD will choose to recall any particular element of His Creation back into service depends on whether it has fully expressed all that GOD intends for it to express about Himself. GOD is love. Where love exists, there remains a story to be told. Whether your 'pet' will ever take on the exact same form again, or finish the 'love story' in another form, forms or as another larger collective expression is all part of the fun and mystery of GOD revealing His nature to us.
You can be sure that if love remains, though, the story is not over. Love never fails, (1 Corinthians 13:8).