Like humans, pets do reincarnate, however, it is rather rare that they do reincarnate during the same lifetime as their human pet parent. The reason for this is not because they do not love us enough to come back, but it is rather that like us humans they have their own purpose or tasks to accomplish while they are in a body. They also have a purpose on the other side. More often than not they may be with us for a certain amount of time that is very important for our growth, or they assist us during times that are rather though. A pet's purpose may be to contribute to their human parent’s spiritual awakening just to provide an example. There is always a reason why they incarnate at a specific time and may or may not reincarnate.
In my experience and by communicating with the other side, is not for the lack of love when they do not come back to us, or when they do not come back instantly. While we miss them in their physical form, they are still with us just without being in a body, therefore they do not miss us the same way we miss them because they are with us at all times and feel the love we have for them.
Reincarnating back into a body can also mean not being able to live a full life because it puts a strain on the body when they come back soon after they left. Therefore, several years on the other side would be ideal. In the spiritual realm there is nothing but unconditional love, which serves as recovery for their soul. All earthly aches and pains are gone, and all they feel is the unconditional love there is on the other side. Yes, they can feel our feelings, including negative feelings, such as anger, anxiety, or sadness, however their essence in the other realm is pure unconditional love, and all they want for us is to be happy. We should always remind ourselves that as difficult as it seems in the moment and especially after losing a beloved pet, loving them unconditionally means putting what is best for their soul’s well-being first.