We need help, suggestions, ideas

My family and I recently rescued what we believe is a 5 year old Russian toy terrier mix. Just had our vet check him out yesterday. He had a shot and then later deworming medicine. Other than being under weight, our new family member seems fine. Our vet agreed that switching our dogs food from kibble to soft would be the best way to gain weight. So this is what we started yesterday. A mix of soft dog food and a bit of kibble. Last night around 2 a.m. the diarrhea started. A 3 hour pause and started again around 6 a.m. and continued for the next couple hours from time to time. Now changing diet was obviously going to cause this reaction but my primary concern is the blood spotting I discovered in his almost translucent, pinkish poop. Our doctor says this is most likely common but if it continues obviously bring him in. So I guess my questions are; has anyone else had a similar experience? Plus. What was the solution? Any help here would be greatly appreciated. Especially by my 8 and 10 year old girls. They/we just love this little guy...