Albeit this being a fairly new phenomenon, I was hoping to find some studies or at least some deep research done on this subject, particularly comparing control groups.
If I were you, I would start with this short statement summarizing the (lack of) evidence and the AAP guidelines for screen exposure during childhood. The references listed at the end will give you more details about the state of knowledge on this topic. Note that there is more research on screen use in general (including computers, movies, and TV) rather than mobile use in particular, so many of the citations in that statement and the others I provide here are not exactly about mobile use. Many of the articles discuss implications for specific kinds of screen exposure, though, so you may still find them helpful.
When considering implications of children's media use, there are two general kinds of questions you can ask:
- Do children experience the intended benefits from media exposure (e.g. do they actually learn from "educational" shows and games?)
- Are there other (unintended) effects of children's media exposure, either short term or long term? Note that these effects could be positive (e.g. better hand eye coordination) or negative (e.g. impaired attention or executive function).
For background information about what children's typical screen exposure is like, here's a large, comprehensive correlational study exploring parent reports on children's media use and outcomes, and another similar study focusing just on very young children (under 2 years)
Do children experience the intended benefits from media exposure?
There are a few controlled experiments available to test this, and most of them by necessity test only immediate effects of screen time rather than longer-term outcomes. For example, this study tests toddlers' ability to learn new words from conversations watched over video compared to in person.
This study tests particular videos explicitly marketed to help infants learn (Baby Einstein videos), showing that infants don't actually learn new words from them without significant scaffolding from caregivers.
(Here's a review that covers several similar studies, if you want to learn more, and another)
There have also been a few studies testing how well infants can learn a foreign language from exposure to media (video recordings), the most famous of which is probably this study showing infants don't learn much about the sound structure of a foreign language unless they get exposed to it live, in person.
Are there other (unintended) effects of children's media exposure?
I don't know of any experimental designs with randomly assigned control groups testing for long term effects of children's media exposure. That kind of study would be extremely difficult to do (parents would need to be VERY invested in adhering to study guidelines over the years necessary to test it) and almost impossible to get ethical approval for, since it potentially puts the participating children at risk for negative outcomes as a result of participating in the study. In the absence of experimental studies, you can look at correlational studies. Longitudinal designs, which measure the same kids over time, can be especially helpful in trying to isolate effects.
This is a study testing the link between screen exposure and later attention problems, but note that it's still not very clear evidence since they couldn't randomly assign children to high or low screen exposure conditions. There could be other variables that contribute to both the amount of screen time and children's attention outcomes, causing an appearance of a causal link between the two even if it's not really there.
Here is an experimental study showing short-term effects of media use (watching a fast-paced children's cartoon) on immediate attention. The authors discuss the impact of such short-term attention impairments and also connect it to research on longer-term attention outcomes and media use.
This is a meta-analysis (a study that statistically combines evidence from lots of other studies) on young children's sedentary activities, such as screen time, and health and cognitive outcomes. This is a review article discussing some of the same questions.
As you point out in your question, this is a young field. The short answer (as pointed out in the first statement I linked to) is that there simply isn't enough evidence right now to confidently make any recommendations for parents one way or another. Unfortunately, this is also a rather emotionally-charged topic, and the weak evidence doesn't stop people (both scientists and lay people) from having very strong opinions. A great approach is to do exactly what you're doing, and try to accumulate as much evidence as you can to help you understand the issue, but acknowledge that there's no easy answer here.